Class 7 ICSE Mathematics: Playing with Numbers – Complete Study Guide

Mathematics is not just about numbers; it’s about understanding patterns, logic, and the relationships between numbers. One of the most interesting chapters in Class 7 ICSE Mathematics is “Playing with Numbers”, which lays the foundation for number theory and helps students develop problem-solving skills. This chapter introduces concepts like factors, multiples, prime numbers, HCF, LCM, divisibility rules, and more. Let’s explore it in depth.

H2: Introduction to Playing with Numbers

“Playing with Numbers” involves understanding and working with numbers in different ways. Here are the main areas covered in this chapter:

Factors and Multiples – How numbers are divisible.
Prime and Composite Numbers – Special types of numbers.
Divisibility Rules – Shortcuts to check if a number is divisible by another.
Highest Common Factor (HCF) – The greatest number that divides two or more numbers.
Lowest Common Multiple (LCM) – The smallest number that is a multiple of two or more numbers.
Tests of Divisibility and Short Tricks – Speed techniques for calculations.

H2: Factors and Multiples

H3: Factors of a Number

A factor is a number that divides another number completely without leaving a remainder.

Example: Factors of 12 are 1, 2, 3, 4, 6, 12.

How to find factors:

Start from 1 and go up to the number itself.
Check which numbers divide it exactly.

Tips: Factors always appear in pairs. For 12:
1 × 12 = 12, 2 × 6 = 12, 3 × 4 = 12

H3: Multiples of a Number

A multiple of a number is obtained by multiplying it with any natural number.

Example: Multiples of 5 are 5, 10, 15, 20, 25…

Important Concept:

Infinite multiples: Every number has infinitely many multiples.
Multiples are used in LCM calculations and divisibility tests.

H3: Prime and Composite Numbers

Prime Number: A number greater than 1 with exactly two factors: 1 and itself.
Example: 2, 3, 5, 7, 11, 13
Composite Number: A number greater than 1 with more than two factors.
Example: 4, 6, 8, 9, 12

Special Cases:

1 is neither prime nor composite.
2 is the smallest prime number and the only even prime number.

Tip: Memorize prime numbers up to 100 for faster calculations.

H3: Tests for Prime Numbers

To check if a number is prime:

Divide it by all prime numbers less than its square root.
If divisible by none, it is prime.

Example: Is 29 prime?

√29 ≈ 5.38 → check 2, 3, 5
29 ÷ 2 → remainder 1
29 ÷ 3 → remainder 2
29 ÷ 5 → remainder 4
✅ Not divisible → 29 is prime

H2: Divisibility Rules

Divisibility rules help check whether a number is divisible by another without performing full division.

H3: Common Divisibility Rules

Divisible by 2: Last digit is even (0, 2, 4, 6, 8)
Divisible by 3: Sum of digits divisible by 3
Divisible by 4: Last two digits divisible by 4
Divisible by 5: Last digit 0 or 5
Divisible by 6: Divisible by both 2 and 3
Divisible by 8: Last three digits divisible by 8
Divisible by 9: Sum of digits divisible by 9
Divisible by 10: Last digit is 0
Divisible by 11: Difference between sum of alternate digits is 0 or divisible by 11

Example: Check 2376 divisibility

Last digit 6 → divisible by 2 ✅
Sum of digits 2+3+7+6=18 → divisible by 3 ✅
Last two digits 76 → divisible by 4 ✅
✅ 2376 divisible by 2, 3, and 4

H2: Highest Common Factor (HCF)

HCF is the largest number that divides two or more numbers exactly.

H3: Methods to Find HCF

Method 1: Prime Factorization

Factorize each number into prime numbers.
Multiply the common prime factors.

Example: HCF of 18 and 24

18 = 2 × 3 × 3
24 = 2 × 2 × 2 × 3
Common primes = 2 × 3 = 6
✅ HCF = 6

Method 2: Division Method

Divide the larger number by the smaller.
Divide the divisor by the remainder repeatedly.
Last divisor is HCF.

Example: HCF of 56 and 42

56 ÷ 42 = 1 remainder 14
42 ÷ 14 = 3 remainder 0
✅ HCF = 14

H2: Lowest Common Multiple (LCM)

LCM is the smallest number divisible by two or more numbers.

H3: Methods to Find LCM

Method 1: Prime Factorization

Factorize numbers into primes.
Take all prime factors with highest powers.
Multiply them.

Example: LCM of 12 and 18

12 = 2² × 3
18 = 2 × 3²
LCM = 2² × 3² = 36

Method 2: Division Method (Short Division)

Write numbers in a row.
Divide by common prime numbers repeatedly.
Multiply all divisors.

Example: LCM of 8 and 12

Divide by 2 → 4, 6
Divide by 2 → 2, 3
Divide by 2 → 1, 3
Divide by 3 → 1, 1
Multiply divisors: 2 × 2 × 2 × 3 = 24 ✅

H3: Relation Between HCF and LCM

For any two numbers a and b:
[
HCF(a, b) × LCM(a, b) = a × b
]

Example: HCF of 12 & 18 = 6, LCM = 36

6 × 36 = 216
12 × 18 = 216 ✅ Correct

Tip: This formula is useful to check answers in exams.

H2: Co-prime Numbers

Co-prime numbers are numbers whose HCF is 1.

Example: 8 and 15 → HCF(8,15)=1 → co-prime

Note: All prime numbers are co-prime with numbers not divisible by them.

H2: Properties of HCF and LCM

HCF of two numbers always ≤ smaller number.
LCM of two numbers always ≥ larger number.
HCF of co-primes = 1
LCM of co-primes = product of numbers

Example: 7 and 15

HCF = 1
LCM = 7 × 15 = 105

H2: Division Algorithm

For any two integers a and b (b ≠ 0), there exist unique integers q and r such that:

[
a = bq + r, \quad 0 ≤ r < |b|
]

a = Dividend
b = Divisor
q = Quotient
r = Remainder

Example: 17 ÷ 5 → 17 = 5×3 + 2

q = 3, r = 2

H2: Tests for Divisibility – Tricks

Divisibility tricks make calculations fast without long division.
Always use sum of digits for 3 and 9.
Alternating sum for 11.
Use last digits for 2, 5, 4, 8, and 10.

Example Trick:

Is 489 divisible by 3?
Sum = 4+8+9 = 21 → divisible by 3 ✅

H2: Interesting Number Patterns

H3: Patterns in Multiplication

Multiplying by 9 → digits sum = 9

9×1=9, 9×2=18 (1+8=9), 9×3=27 (2+7=9)

Squares of odd numbers → end with 1, 9, 5, 5, 9, 1 (cyclic)

H3: Magic of HCF and LCM

Product of HCF and LCM = Product of numbers
HCF always divides numbers exactly
LCM always divisible by numbers

H3: Special Properties

If numbers are consecutive integers, HCF=1
If numbers are multiples of a common number, HCF ≥ 1

H2: Real-Life Applications

HCF: Dividing items into equal groups.

Example: 12 pencils and 18 pens → largest equal group = HCF = 6

LCM: Planning events at common intervals.

Example: Two traffic lights flashing every 12 sec & 18 sec → sync after LCM = 36 sec

Divisibility Rules: Quick calculations, error checking in exams.

H2: Solved Examples

Find HCF and LCM of 24 and 36

24 = 2³ × 3
36 = 2² × 3²
HCF = 2² × 3 = 12
LCM = 2³ × 3² = 72

Check divisibility of 351 by 9

Sum = 3+5+1=9 → divisible ✅

Find co-primes among 15, 28, 35

HCF(15,28)=1 → co-prime
HCF(15,35)=5 → not co-prime
HCF(28,35)=7 → not co-prime

H2: Practice Questions

List all factors of 36.
Find LCM and HCF of 20 and 30.
Determine if 121 is divisible by 11.
Check which pairs are co-prime: (14,15), (21,28)
Solve using division algorithm: 59 ÷ 7

Tips: Practice these regularly to build speed and accuracy.

H2: Summary

Factors: Divide numbers exactly
Multiples: Numbers obtained by multiplying
Prime Numbers: Only 1 and itself as factors
Composite Numbers: More than 2 factors
HCF: Greatest common divisor
LCM: Smallest common multiple
Divisibility Rules: Shortcuts for checking divisibility
Co-prime Numbers: HCF = 1
Division Algorithm: Express dividend as divisor × quotient + remainder

Mastering these concepts helps students solve complex number problems quickly and builds a strong foundation for algebra, fractions, and higher mathematics.

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Class 7 ICSE Mathematics – Playing with Numbers (Easy Notes)
(As per the syllabus of the Council for the Indian School Certificate Examinations)
Introduction
The chapter “Playing with Numbers” in Class 7 ICSE Mathematics is an important foundation chapter. It helps students understand how numbers behave, how they can be divided, and how certain rules apply to numbers. This chapter improves logical thinking and builds a strong base for higher classes.
In this chapter, students learn about:
Factors and multiples
Divisibility rules
Prime and composite numbers
Tests of divisibility
HCF (Highest Common Factor)
LCM (Least Common Multiple)
Properties of numbers
These concepts are useful in daily life, competitive exams, and advanced mathematics.

Natural Numbers
Natural numbers are the numbers we use for counting.
Definition
Natural numbers are positive whole numbers starting from 1.
Examples
1, 2, 3, 4, 5, 6, …
Properties
Smallest natural number is 1
Natural numbers are infinite
Zero (0) is not a natural number
Whole Numbers
Whole numbers include all natural numbers and zero.
Definition
Whole numbers are numbers starting from 0 and going up to infinity.
Examples
0, 1, 2, 3, 4, 5, …
Properties
Smallest whole number is 0
No largest whole number
All natural numbers are whole numbers
Factors
Factors are numbers that divide another number exactly without leaving any remainder.
Definition
A factor of a number is a number that divides it completely.
Example
Factors of 12 are:
1, 2, 3, 4, 6, 12
Important Points
1 is a factor of every number
Every number is a factor of itself
Factors are always less than or equal to the number
Multiples
Multiples are numbers obtained by multiplying a given number by natural numbers.
Definition
A multiple of a number is the result when the number is multiplied by 1, 2, 3, 4, etc.
Example
Multiples of 5 are:
5, 10, 15, 20, 25, …
Important Points
Multiples are greater than or equal to the number
Every number has infinite multiples
Prime Numbers
Prime numbers play a very important role in mathematics.
Definition
A prime number is a number greater than 1 that has only two factors:
1 and itself.
Examples
2, 3, 5, 7, 11, 13, 17, 19
Important Facts
2 is the smallest prime number
2 is the only even prime number
1 is not a prime number
Composite Numbers
Composite numbers are numbers that have more than two factors.
Definition
A composite number is a number that has more than two factors.
Examples
4, 6, 8, 9, 10, 12, 15
Comparison: Prime vs Composite
Prime Number
Composite Number
Has exactly 2 factors
Has more than 2 factors
Example: 7
Example: 9
Co-prime Numbers
Two numbers are called co-prime if they have no common factor other than 1.
Examples
8 and 15 (common factor is only 1)
9 and 20
Important Note
Co-prime numbers do not need to be prime individually.
Divisibility Rules
Divisibility rules help us check whether a number is divisible by another number without actual division.
Divisibility by 2
A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).
Example:
246 is divisible by 2
Divisibility by 3
A number is divisible by 3 if the sum of its digits is divisible by 3.
Example:
Sum of digits of 132 = 1 + 3 + 2 = 6
Since 6 is divisible by 3, 132 is divisible by 3
Divisibility by 4
A number is divisible by 4 if the last two digits are divisible by 4.
Example:
316 → last two digits = 16
16 ÷ 4 = 4, so divisible by 4
Divisibility by 5
A number is divisible by 5 if it ends in 0 or 5.
Example:
125 ends in 5 → divisible by 5
Divisibility by 6
A number is divisible by 6 if it is divisible by both 2 and 3.
Example:
132 is divisible by 2 and 3 → divisible by 6
Divisibility by 8
A number is divisible by 8 if the last three digits are divisible by 8.
Example:
1016 → last three digits = 016 = 16
16 ÷ 8 = 2 → divisible by 8
Divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.
Example:
729 → 7 + 2 + 9 = 18
18 is divisible by 9 → number is divisible by 9
Divisibility by 10
A number is divisible by 10 if it ends in 0.
Example:
450 ends in 0 → divisible by 10
Even and Odd Numbers
Even Numbers
Numbers divisible by 2 are even numbers.
Examples:
2, 4, 6, 8, 10
Odd Numbers
Numbers not divisible by 2 are odd numbers.
Examples:
1, 3, 5, 7, 9
Properties
Even + Even = Even
Odd + Odd = Even
Even + Odd = Odd
Highest Common Factor (HCF)
HCF is the greatest number that divides two or more numbers completely.
Methods to Find HCF
Listing factors
Prime factorisation
Division method
Example
Find HCF of 12 and 18.
Factors of 12 = 1, 2, 3, 4, 6, 12
Factors of 18 = 1, 2, 3, 6, 9, 18
Common factors = 1, 2, 3, 6
HCF = 6
Least Common Multiple (LCM)
LCM is the smallest number that is a multiple of two or more numbers.
Methods to Find LCM
Listing multiples
Prime factorisation
Division method
Example
Find LCM of 4 and 6.
Multiples of 4: 4, 8, 12, 16, …
Multiples of 6: 6, 12, 18, …
LCM = 12
Relationship between HCF and LCM
For two numbers:
Example
Numbers: 8 and 12
HCF = 4
LCM = 24
4 × 24 = 96
8 × 12 = 96 ✔
Use of Playing with Numbers in Daily Life
Finding equal distribution (HCF)
Planning events and schedules (LCM)
Checking number patterns
Solving puzzles and logical problems
Useful in competitive exams
Important Tips for Students
Learn divisibility rules by heart
Practice factorisation regularly
Understand concepts instead of memorising
Solve word problems carefully
Revise tables up to 20
Summary
In the chapter “Playing with Numbers”, we learned:
Types of numbers
Factors and multiples
Prime and composite numbers
Divisibility rules
HCF and LCM
Properties of numbers
This chapter strengthens the foundation of mathematics and prepares students for advanced topics in higher classes.
✅ Revision Advice
Practice sums daily, revise divisibility tests, and solve previous years’ ICSE questions for better understanding.
If you want:
Solved examples
Practice worksheets
MCQs
Extra word problems
Just tell me! 😊

Natural Numbers
Natural numbers are the numbers we use for counting.
Definition
Natural numbers are positive whole numbers starting from 1.
Examples
1, 2, 3, 4, 5, 6, …
Properties
Smallest natural number is 1
Natural numbers are infinite
Zero (0) is not a natural number
Whole Numbers
Whole numbers include all natural numbers and zero.
Definition
Whole numbers are numbers starting from 0 and going up to infinity.
Examples
0, 1, 2, 3, 4, 5, …
Properties
Smallest whole number is 0
No largest whole number
All natural numbers are whole numbers
Factors
Factors are numbers that divide another number exactly without leaving any remainder.
Definition
A factor of a number is a number that divides it completely.
Example
Factors of 12 are:
1, 2, 3, 4, 6, 12
Important Points
1 is a factor of every number
Every number is a factor of itself
Factors are always less than or equal to the number
Multiples
Multiples are numbers obtained by multiplying a given number by natural numbers.
Definition
A multiple of a number is the result when the number is multiplied by 1, 2, 3, 4, etc.
Example
Multiples of 5 are:
5, 10, 15, 20, 25, …
Important Points
Multiples are greater than or equal to the number
Every number has infinite multiples
Prime Numbers
Prime numbers play a very important role in mathematics.
Definition
A prime number is a number greater than 1 that has only two factors:
1 and itself.
Examples
2, 3, 5, 7, 11, 13, 17, 19
Important Facts
2 is the smallest prime number
2 is the only even prime number
1 is not a prime number
Composite Numbers
Composite numbers are numbers that have more than two factors.
Definition
A composite number is a number that has more than two factors.
Examples
4, 6, 8, 9, 10, 12, 15
Comparison: Prime vs Composite
Prime Number
Composite Number
Has exactly 2 factors
Has more than 2 factors
Example: 7
Example: 9
Co-prime Numbers
Two numbers are called co-prime if they have no common factor other than 1.
Examples
8 and 15 (common factor is only 1)
9 and 20
Important Note
Co-prime numbers do not need to be prime individually.
Divisibility Rules
Divisibility rules help us check whether a number is divisible by another number without actual division.
Divisibility by 2
A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).
Example:
246 is divisible by 2
Divisibility by 3
A number is divisible by 3 if the sum of its digits is divisible by 3.
Example:
Sum of digits of 132 = 1 + 3 + 2 = 6
Since 6 is divisible by 3, 132 is divisible by 3
Divisibility by 4
A number is divisible by 4 if the last two digits are divisible by 4.
Example:
316 → last two digits = 16
16 ÷ 4 = 4, so divisible by 4
Divisibility by 5
A number is divisible by 5 if it ends in 0 or 5.
Example:
125 ends in 5 → divisible by 5
Divisibility by 6
A number is divisible by 6 if it is divisible by both 2 and 3.
Example:
132 is divisible by 2 and 3 → divisible by 6
Divisibility by 8
A number is divisible by 8 if the last three digits are divisible by 8.
Example:
1016 → last three digits = 016 = 16
16 ÷ 8 = 2 → divisible by 8
Divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.
Example:
729 → 7 + 2 + 9 = 18
18 is divisible by 9 → number is divisible by 9
Divisibility by 10
A number is divisible by 10 if it ends in 0.
Example:
450 ends in 0 → divisible by 10
Even and Odd Numbers
Even Numbers
Numbers divisible by 2 are even numbers.
Examples:
2, 4, 6, 8, 10
Odd Numbers
Numbers not divisible by 2 are odd numbers.
Examples:
1, 3, 5, 7, 9
Properties
Even + Even = Even
Odd + Odd = Even
Even + Odd = Odd
Highest Common Factor (HCF)
HCF is the greatest number that divides two or more numbers completely.
Methods to Find HCF
Listing factors
Prime factorisation
Division method
Example
Find HCF of 12 and 18.
Factors of 12 = 1, 2, 3, 4, 6, 12
Factors of 18 = 1, 2, 3, 6, 9, 18
Common factors = 1, 2, 3, 6
HCF = 6
Least Common Multiple (LCM)
LCM is the smallest number that is a multiple of two or more numbers.
Methods to Find LCM
Listing multiples
Prime factorisation
Division method
Example
Find LCM of 4 and 6.
Multiples of 4: 4, 8, 12, 16, …
Multiples of 6: 6, 12, 18, …
LCM = 12
Relationship between HCF and LCM
For two numbers:
Example
Numbers: 8 and 12
HCF = 4
LCM = 24
4 × 24 = 96
8 × 12 = 96 ✔
Use of Playing with Numbers in Daily Life
Finding equal distribution (HCF)
Planning events and schedules (LCM)
Checking number patterns
Solving puzzles and logical problems
Useful in competitive exams
Important Tips for Students
Learn divisibility rules by heart
Practice factorisation regularly
Understand concepts instead of memorising
Solve word problems carefully
Revise tables up to 20
Summary
In the chapter “Playing with Numbers”, we learned:
Types of numbers
Factors and multiples
Prime and composite numbers
Divisibility rules
HCF and LCM
Properties of numbers
This chapter strengthens the foundation of mathematics and prepares students for advanced topics in higher classes.
✅ Revision Advice
Practice sums daily, revise divisibility tests, and solve previous years’ ICSE questions for better understanding.
If you want:
Solved examples
Practice worksheets
MCQs
Extra word problems
Just tell me! 😊

Below is MORE continuation, written in the same simple ICSE-friendly language, and directly extendable to your existing notes. You can paste it after your last section without editing.

Prime Factorisation
Prime factorisation is one of the most important ideas in the chapter Playing with Numbers. It helps us find HCF, LCM, and simplify problems easily.
Definition
Prime factorisation is the process of expressing a number as a product of only prime numbers.
Example 1
Prime factorisation of 36:
36 = 2 × 18
18 = 2 × 9
9 = 3 × 3
So,
36 = 2 × 2 × 3 × 3
36 = 2² × 3²
Example 2
Prime factorisation of 84:
84 = 2 × 42
42 = 2 × 21
21 = 3 × 7
So,
84 = 2² × 3 × 7
Why Prime Factorisation is Important
Helps in finding HCF
Helps in finding LCM
Used in simplifying fractions
Makes calculations easier in exams
Finding HCF Using Prime Factorisation
Steps
Write prime factorisation of each number
Identify common prime factors
Take the lowest power of each common factor
Multiply them
Example
Find HCF of 24 and 36
24 = 2³ × 3
36 = 2² × 3²
Common factors = 2² × 3
HCF = 12
Finding LCM Using Prime Factorisation
Steps
Write prime factorisation of the numbers
Take highest power of each prime factor
Multiply them
Example
Find LCM of 12 and 18
12 = 2² × 3
18 = 2 × 3²
LCM = 2² × 3²
LCM = 36
HCF and LCM Word Problems
Example 1
Find the largest number that divides 36 and 60 exactly.
36 = 2² × 3²
60 = 2² × 3 × 5
HCF = 2² × 3 = 12
Example 2
Find the smallest number divisible by 6, 8 and 12.
6 = 2 × 3
8 = 2³
12 = 2² × 3
LCM = 2³ × 3 = 24
Properties of Prime Numbers
Every prime number greater than 1 has exactly two factors
2 is the only even prime number
There are infinitely many prime numbers
Prime numbers greater than 3 are of the form 6n ± 1
Twin Prime Numbers
Definition
Two prime numbers whose difference is 2 are called twin primes.
Examples
(3, 5)
(11, 13)
(17, 19)
Even and Odd Numbers – More Properties
Addition
Even + Even = Even
Odd + Odd = Even
Even + Odd = Odd
Multiplication
Even × Even = Even
Odd × Odd = Odd
Even × Odd = Even
Divisibility Rules – Quick Revision Table
Number
Rule
2
Last digit is even
3
Sum of digits divisible by 3
4
Last two digits divisible by 4
5
Ends in 0 or 5
6
Divisible by 2 and 3
8
Last three digits divisible by 8
9
Sum of digits divisible by 9
10
Ends in 0
Common Mistakes to Avoid
Thinking 1 is a prime number ❌
Using highest power in HCF ❌
Using lowest power in LCM ❌
Confusing factors with multiples
Skipping steps in prime factorisation
Mental Maths Tips
Learn tables up to 20
Revise divisibility rules daily
Practice factor trees
Always check answers
Write steps clearly in exams
Multiple Choice Questions (MCQs)
Which of the following is a composite number?
a) 2
b) 5
c) 9
d) 11
Answer: c) 9
HCF of 18 and 27 is
a) 3
b) 6
c) 9
d) 12
Answer: c) 9
LCM of 4 and 10 is
a) 20
b) 40
c) 10
d) 5
Answer: a) 20
Very Short Answer Questions
Smallest composite number → 4
HCF of co-prime numbers → 1
Only even prime number → 2
Factor of every number → 1
Multiple of every number → 0
Short Answer Questions
Define prime factorisation
What are co-prime numbers?
Write any two divisibility rules
Define HCF
Define LCM
Long Answer Questions
Explain prime and composite numbers with examples
Find HCF and LCM using prime factorisation
Write properties of even and odd numbers
Explain divisibility rules of 3 and 9
Solve word problems based on HCF and LCM
Importance of Playing with Numbers
Builds strong number sense
Improves logical thinking
Useful for competitive exams
Foundation for algebra and number theory
Important as per the syllabus of the Council for the Indian School Certificate Examinations
Final Conclusion
The chapter Playing with Numbers is one of the most important chapters of Class 7 ICSE Mathematics. A strong understanding of this chapter helps students perform better in exams and prepares them for higher classes.
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Worksheet (PDF-style content)
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Numbers on the Number Line
A number line is a straight line on which numbers are represented at equal intervals.
Key Points
Numbers increase as we move to the right
Numbers decrease as we move to the left
Zero lies at the centre of the number line
Importance
Helps in comparing numbers
Useful in understanding addition and subtraction
Builds clarity of number order
Comparing Numbers
Rules for Comparing Numbers
A number with more digits is always greater
If digits are equal, compare place values from left to right
Example
Compare 567 and 589
Hundreds digit same (5)
Tens digit: 6 < 8 So, 589 > 567
Ascending and Descending Order
Ascending Order
Numbers arranged from smallest to greatest
Example:
3, 7, 12, 19
Descending Order
Numbers arranged from greatest to smallest
Example:
25, 18, 10, 4
Test of Divisibility – Application Based Questions
Example 1
Check whether 540 is divisible by 9.
Sum of digits = 5 + 4 + 0 = 9
Since 9 is divisible by 9,
540 is divisible by 9
Example 2
Check divisibility of 728 by 8.
Last three digits = 728
728 ÷ 8 = 91
So, 728 is divisible by 8
Use of HCF in Real Life
HCF is used when we want to divide things equally without leftovers.
Example
48 chocolates and 72 biscuits are to be distributed equally among students.
Maximum number of students = HCF of 48 and 72
48 = 2⁴ × 3
72 = 2³ × 3²
HCF = 2³ × 3 = 24
So, chocolates and biscuits can be distributed among 24 students.
Use of LCM in Real Life
LCM is used when events repeat after fixed intervals.
Example
Two bells ring every 6 minutes and 8 minutes.
When will they ring together again?
LCM of 6 and 8 = 24
So, bells will ring together after 24 minutes.
Difference Between HCF and LCM
HCF
LCM
Greatest common factor
Smallest common multiple
Used for equal distribution
Used for scheduling
Lowest power of primes
Highest power of primes
HOTS (Higher Order Thinking Skills) Questions
1.
If HCF of two numbers is 1, what are they called?
Answer: Co-prime numbers
2.
Can two prime numbers have HCF greater than 1?
Answer: No, because prime numbers have only two factors
3.
If one number is a multiple of another, what is their HCF?
Answer: The smaller number
Assertion and Reason Questions
Q1
Assertion (A): 1 is neither prime nor composite
Reason (R): 1 has only one factor
Correct Answer: Both A and R are true
Fill in the Blanks
The smallest prime number is _ Answer: 2 The HCF of two co-prime numbers is
Answer: 1
A number divisible by 10 ends in __
Answer: 0
True or False
All even numbers are composite ❌
2 is the only even prime number ✔
Multiples of a number are finite ❌
Every number has at least two factors ❌
Match the Following
Column A
Column B
Prime number
Has two factors
Composite number
Has more than two factors
HCF
Greatest factor
LCM
Smallest multiple
Case Study Based Question (ICSE Pattern)
A school has 36 boys and 48 girls. The principal wants to arrange them in equal rows.
What is the maximum number of students in each row?
Which mathematical concept is used?
Solution:
HCF of 36 and 48 = 12
Concept used: HCF
One-Mark Questions (Exam Ready)
Define a prime number
What is the smallest whole number?
Write one multiple of 9
What is the HCF of 5 and 7?
Two-Mark Questions
Write any two divisibility rules
Give two examples of composite numbers
Define co-prime numbers with example
Five-Mark Questions
Explain divisibility rules of 2, 3 and 5
Find HCF and LCM using prime factorisation
Differentiate between factors and multiples
Common Exam Mistakes (ICSE Warning ⚠️)
Writing 1 as a prime number
Skipping factorisation steps
Mixing up HCF and LCM
Not showing working properly
Quick Revision Chart
Prime → 2 factors
Composite → More than 2 factors
HCF → Greatest factor
LCM → Smallest multiple
Final Exam Strategy
Revise divisibility rules daily
Practice word problems
Write steps clearly
Check answers logically
Follow ICSE marking scheme of the Council for the Indian School Certificate Examinations
Ultimate Conclusion
The chapter Playing with Numbers builds the foundation of mathematics. Mastering this chapter ensures strong performance in Class 7 ICSE exams and prepares students for higher classes like Class 8 and 9.
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Factors and Multiples – Deep Understanding
Students often confuse factors and multiples, so this section explains them clearly.
Key Differences
Factors
Multiples
Divide the number exactly
Obtained by multiplication
Finite in number
Infinite in number
Always ≤ the number
Always ≥ the number
Example
For number 12:
Factors: 1, 2, 3, 4, 6, 12
Multiples: 12, 24, 36, 48, …
Common Factors and Common Multiples
Common Factors
Factors that are common to two or more numbers.
Example:
Factors of 18 = 1, 2, 3, 6, 9, 18
Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24
Common factors = 1, 2, 3, 6
Common Multiples
Multiples common to two or more numbers.
Example:
Multiples of 4 = 4, 8, 12, 16, 20, …
Multiples of 6 = 6, 12, 18, 24, …
Common multiples = 12, 24, …
Smallest common multiple = LCM
HCF and LCM Using Division Method
Steps
Divide the larger number by the smaller
Continue division until remainder becomes 0
The last divisor is the HCF
Example
Find HCF of 20 and 28
28 ÷ 20 = 1 remainder 8
20 ÷ 8 = 2 remainder 4
8 ÷ 4 = 2 remainder 0
HCF = 4
LCM Using Division Method
Example
Find LCM of 8, 12 and 16
Divide by common prime numbers:
2 | 8, 12, 16
2 | 4, 6, 8
2 | 2, 3, 4
2 | 1, 3, 2
3 | 1, 3, 1
= 2 × 2 × 2 × 2 × 3 = 48
Special Results in Playing with Numbers
HCF of two consecutive numbers = 1
HCF of two prime numbers = 1
LCM of co-prime numbers = Product of numbers
HCF of a number with itself = The number
Numbers Having Exactly Two Factors
Numbers with exactly two factors are called prime numbers.
Why 1 is Not Prime
1 has only one factor
Prime numbers must have two factors
Pattern Questions (ICSE Favourite)
Example
Find the next number in the pattern:
2, 4, 8, 16, _
Each number is multiplied by 2
Next number = 32
Logical Reasoning Questions
Q
Can a composite number be even?
Answer:
Yes. Example: 4, 6, 8 are even composite numbers.
Word Problems – Mixed Practice
Problem
Three ropes of length 12 m, 18 m and 24 m are to be cut into equal pieces of maximum length.
Solution:
HCF of 12, 18 and 24
12 = 2² × 3
18 = 2 × 3²
24 = 2³ × 3
HCF = 2 × 3 = 6 m
Step-by-Step Exam Writing Format (Very Important)
In ICSE exams:
Write Given
Write To Find
Write Solution
Write Answer clearly
This helps in getting full marks.
Self-Assessment Questions
Is 91 a prime number?
Find HCF of 15 and 20
Find LCM of 3, 5 and 7
Write first five multiples of 9
Revision Worksheet (Practice Set)
A. Find
Factors of 36
Multiples of 7 (first five)
B. Find HCF
24 and 60
18 and 45
C. Find LCM
6 and 15
8 and 20
Value-Based Question
Why is learning divisibility rules important?
Answer:
They help save time, improve mental maths, and make calculations easier in daily life and exams.
Student-Friendly Summary
Factors divide numbers
Multiples are products
Prime numbers have two factors
Composite numbers have many factors
HCF is the greatest factor
LCM is the smallest multiple
Final One-Page Memory Points
Smallest prime = 2
Smallest composite = 4
HCF of co-primes = 1
LCM of co-primes = Product
1 is neither prime nor composite
Absolute Final Conclusion
The chapter Playing with Numbers is the heart of number theory at the middle-school level. A strong command over this chapter ensures success not only in Class 7 ICSE, but also builds confidence for higher classes and competitive exams.
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Understanding Factors Using Factor Pairs
A factor pair is a pair of numbers that multiply to give the original number.
Example
Factor pairs of 24:
1 × 24
2 × 12
3 × 8
4 × 6
👉 From factor pairs, we can easily list all factors.
Identifying Prime and Composite Numbers Quickly
Quick Test
If a number has only one factor pair, it is prime
If it has more than one factor pair, it is composite
Example
13 → (1, 13) → Prime
18 → (1,18), (2,9), (3,6) → Composite
Special Types of Numbers
Perfect Numbers
A number is called a perfect number if the sum of its proper factors equals the number itself.
Example:
6 → Proper factors: 1, 2, 3
1 + 2 + 3 = 6 ✔
Square Numbers
Numbers that are obtained by multiplying a number by itself.
Examples:
1, 4, 9, 16, 25
Cube Numbers
Numbers obtained by multiplying a number three times.
Examples:
1, 8, 27, 64
Use of Divisibility Rules in Exams
Divisibility rules help:
Save time
Avoid long division
Improve accuracy
ICSE Tip
Always mention the rule used while solving divisibility questions to get full marks.
Why 0 Is Not Considered a Factor
A factor must divide a number
Division by 0 is not defined
Hence, 0 is never a factor
Zero and Its Properties
0 is a whole number
0 is neither positive nor negative
0 has no reciprocal
Any number multiplied by 0 = 0
More Word Problems on HCF
Example
Three containers hold 20 L, 30 L and 50 L of milk. Milk is to be poured equally into bottles.
Maximum capacity of each bottle = HCF of 20, 30 and 50
20 = 2² × 5
30 = 2 × 3 × 5
50 = 2 × 5²
HCF = 2 × 5 = 10 L
More Word Problems on LCM
Example
Three traffic lights blink after every 15 s, 20 s and 30 s.
When will they blink together again?
LCM of 15, 20, 30 = 60 seconds
Difference Between Prime Factorisation and Division Method
Prime Factorisation
Division Method
Uses factor trees
Uses continuous division
Easy to understand
Faster in exams
Best for beginners
Best for large numbers
Trick to Remember HCF and LCM
👉 HCF → Divide equally
👉 LCM → Repeat together
Real-Life Examples
Packing items → HCF
Alarm clocks → LCM
Seating arrangements → HCF
Timetables → LCM
Exam-Oriented Practice Questions
A. One Mark
Write the smallest composite number
Write one multiple of 12
B. Two Marks
Define co-prime numbers with example
Write divisibility rules of 2 and 5
C. Four Marks
Find HCF of 24 and 36
Find LCM of 6, 8 and 12
Reasoning Questions
Q
Why is every prime number greater than 2 odd?
Answer:
Because any even number greater than 2 is divisible by 2 and hence composite.
Mental Maths Challenge
Find HCF of 9 and 10 (Answer: 1)
Find LCM of 4 and 5 (Answer: 20)
Is 49 prime? (No)
Common Student Doubts (Clarified)
❓ Is 1 prime? → ❌ No
❓ Can two numbers have more than one HCF? → ❌ No
❓ Can LCM be smaller than numbers? → ❌ No
Flow Chart (For Revision)
Number →
Check factors →
Two factors → Prime
More than two → Composite
Last-Minute Revision Points
Learn divisibility rules
Practice factorisation
Understand word problems
Write steps clearly
Revise formulas
Teacher’s Remark Section (For Notebook)
“Good understanding of number concepts. Needs more practice in word problems.”
Ultra-Short Summary
Numbers follow rules
Factors divide
Multiples repeat
Prime numbers are special
HCF and LCM solve real problems
FINAL TAKEAWAY
Mastering Playing with Numbers gives students confidence, speed, and accuracy in mathematics. This chapter is a pillar chapter for success in Class 7 ICSE and higher classes.
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Understanding Divisibility Through Examples
Divisibility rules become clearer when we apply them to real numbers.
Example 1
Check whether 1,248 is divisible by 4.
Last two digits = 48
48 ÷ 4 = 12
✔ The number is divisible by 4.
Example 2
Check whether 3,456 is divisible by 9.
Sum of digits = 3 + 4 + 5 + 6 = 18
18 is divisible by 9
✔ The number is divisible by 9.
Why Divisibility Rules Are Important
Reduce calculation time in exams
Help in mental maths
Used in HCF and LCM problems
Increase accuracy
👉 ICSE Tip: Always mention the divisibility rule used.
Composite Numbers and Their Properties
All composite numbers have more than two factors
All even numbers except 2 are composite
Composite numbers can be odd or even
Examples
Even composite: 4, 6, 8
Odd composite: 9, 15, 21
Prime Numbers Between Two Numbers
Example
Find prime numbers between 10 and 20.
Numbers: 11, 13, 17, 19
✔ All are prime because they have exactly two factors.
Finding Factors Using Division Method
Example
Find factors of 18.
18 ÷ 1 = 18
18 ÷ 2 = 9
18 ÷ 3 = 6
So, factors are 1, 2, 3, 6, 9, 18
Relationship Between Factors and Multiples
If a is a factor of b, then b is a multiple of a
Factors divide, multiples multiply
Example
3 is a factor of 15
15 is a multiple of 3
Common Factors vs Common Multiples
Common Factors
Common Multiples
Used to find HCF
Used to find LCM
Finite
Infinite
Smaller numbers
Larger numbers
More HOTS Questions
Q1
If HCF of two numbers is equal to one of the numbers, what can you say?
Answer:
The smaller number divides the larger number exactly.
Q2
Can the LCM of two numbers be equal to one of the numbers?
Answer:
Yes, when one number is a multiple of the other.
Case Study Question (Exam Pattern)
Two numbers have HCF = 5 and LCM = 100.
Find the product of the two numbers.
Solution:
HCF × LCM = Product of numbers
5 × 100 = 500
Mathematical Vocabulary (Important Terms)
Factor – A number that divides another exactly
Multiple – Result of multiplication
Prime – Exactly two factors
Composite – More than two factors
Co-prime – Only common factor is 1
Error Analysis (Learn From Mistakes)
❌ Writing 1 as a prime number
❌ Missing common factors
❌ Taking wrong power in LCM
❌ Not showing steps
✔ Always double-check.
Speed Maths Tips
Use divisibility rules instead of division
Memorise prime numbers up to 50
Write rough work neatly
Eliminate wrong MCQ options quickly
Full Chapter Recap
Numbers follow fixed rules
Factors divide numbers
Multiples are infinite
Prime numbers have two factors
HCF helps in equal division
LCM helps in repetition problems
Student Self-Check List
✔ Can I find HCF by all methods?
✔ Can I apply divisibility rules correctly?
✔ Can I solve word problems?
✔ Do I remember properties of numbers?
Final Motivation for Students
If you understand Playing with Numbers, mathematics becomes easy and interesting. This chapter is the base of number theory, and mastering it ensures strong performance in Class 7 ICSE exams and confidence in higher classes.
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Understanding Numbers Through Real-Life Situations
Numbers are not only used in textbooks but also in daily life.
Examples
Counting students in a class → Natural numbers
Dividing sweets equally → Factors and HCF
Setting alarms and schedules → Multiples and LCM
This chapter helps students apply maths in real situations.
Why Prime Numbers Are Important
Prime numbers are called the building blocks of numbers.
Reasons
Every number can be written as a product of prime numbers
Used in cryptography and computer science
Important for higher mathematics
Checking Whether a Number Is Prime (Method)
To check whether a number is prime:
Try dividing it by prime numbers only
Stop checking after the square root of the number
Example
Check whether 29 is prime.
Try dividing by 2, 3, 5
29 is not divisible by any
✔ 29 is a prime number
Numbers Having Exactly Three Factors
A number having exactly three factors is always a square of a prime number.
Examples
4 = 2² → Factors: 1, 2, 4
9 = 3² → Factors: 1, 3, 9
Factors of 1 and 0
1 has only one factor → 1
0 has infinitely many factors because every number divides 0
👉 This is why 0 is neither prime nor composite.
More Practice on Factors
Example
Find the number of factors of 16.
16 = 2⁴
Number of factors = (4 + 1) = 5
Factors: 1, 2, 4, 8, 16
Finding LCM Using Formula
For two numbers:
Example
Numbers: 12 and 18
HCF = 6
LCM = (12 × 18) ÷ 6 = 36
Mixed Concept Questions
Q1
Find HCF and LCM of 15 and 25.
15 = 3 × 5
25 = 5 × 5
HCF = 5
LCM = 75
Application-Based Question
A farmer has 18 mangoes and 24 apples. He wants to distribute them equally.
Which concept is used? → HCF
Answer: 6 baskets
Important ICSE Keywords
“Greatest number that divides” → HCF
“Smallest number divisible by” → LCM
“Exactly divisible” → Factor
“Repeated after equal intervals” → LCM
Stepwise Presentation (Marks Booster)
Always write:
Prime factorisation clearly
Final answer in a box
Units (if any)
This improves presentation and marks.
Self-Practice Drill
A. Find HCF
21 and 28
16 and 40
B. Find LCM
9 and 12
10 and 15
Mental Ability Questions
What is the HCF of two consecutive numbers? → 1
Is 51 divisible by 3? → Yes (5 + 1 = 6)
Common Confusions Cleared
Prime ≠ Odd (example: 2)
Composite ≠ Even (example: 9)
Bigger number ≠ Bigger HCF
Exam-Time Strategy
Read the question carefully
Identify whether HCF or LCM is needed
Choose the correct method
Show steps clearly
Rapid Revision Table
Concept
Key Idea
Factor
Divides exactly
Multiple
Product
Prime
2 factors
Composite
More than 2 factors
HCF
Greatest factor
LCM
Smallest multiple
Confidence Builder
If you can: ✔ Apply divisibility rules
✔ Find HCF & LCM correctly
✔ Solve word problems
Then you have mastered this chapter.
Absolute Final Wrap-Up
The chapter Playing with Numbers is a foundation chapter in Class 7 ICSE Mathematics. It strengthens logical thinking, improves calculation speed, and prepares students for advanced topics like algebra and number theory.
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Finding Whether a Number Is a Perfect Square (Basic Idea)
A perfect square is a number obtained by multiplying a number by itself.
Quick Check Using Prime Factorisation
If all prime factors occur in pairs, the number is a perfect square.
Example
Check whether 144 is a perfect square.
144 = 2⁴ × 3²
All powers are even ✔
So, 144 is a perfect square
Numbers With Exactly Four Factors
A number having exactly four factors can be:
Product of two distinct prime numbers, or
Cube of a prime number
Examples
6 = 2 × 3 → Factors: 1, 2, 3, 6
8 = 2³ → Factors: 1, 2, 4, 8
Counting Factors (Simple Method)
If a number is written as
Example
For 27 = 3³
Number of factors = 3 + 1 = 4
Counting Factors (Two Prime Factors)
If
Example
For 36 = 2² × 3²
Number of factors = (2 + 1)(2 + 1) = 9
Why Prime Numbers Have Only Two Factors
One factor is 1
The other factor is the number itself
No other number divides it exactly
This makes prime numbers special.
Finding the Greatest Prime Factor
Example
Find the greatest prime factor of 90.
90 = 2 × 3² × 5
Greatest prime factor = 5
Finding the Least Prime Factor
Example
Find the least prime factor of 91.
91 = 7 × 13
Least prime factor = 7
ICSE-Type Reasoning Questions
Q
Why is 2 not considered a composite number?
Answer:
Because it has exactly two factors, not more than two.
More Word Problems (Mixed Concepts)
Problem
Find the smallest number which when divided by 12, 15 and 18 leaves no remainder.
This is an LCM problem.
12 = 2² × 3
15 = 3 × 5
18 = 2 × 3²
LCM = 2² × 3² × 5 = 180
Equal Remainder Type Question
Problem
Find the largest number that divides 45, 75 and 105 leaving the same remainder.
Find HCF of differences:
75 − 45 = 30
105 − 75 = 30
HCF of 30 and 30 = 30
Trick to Identify HCF or LCM in Word Problems
Words like “maximum”, “greatest”, “largest number” → HCF
Words like “smallest”, “together again”, “least number” → LCM
Revision Drill (Speed Practice)
HCF of 14 and 21 → 7
LCM of 7 and 9 → 63
Is 37 prime? → Yes
Write one multiple of 15 → 30
Assertion–Reason (Practice)
Assertion (A): Every composite number has a prime factor.
Reason (R): Composite numbers can be expressed as a product of primes.
Answer: Both A and R are true.
Mistakes That Reduce Marks
Not boxing the final answer
Skipping steps
Wrong method selection
Poor presentation
👉 Neat work = better marks.
Chapter-Based Oral Questions
What is the smallest prime number?
Define co-prime numbers.
Give one example of a composite number.
What is the HCF of two prime numbers?
Student Confidence Checklist
✔ I know divisibility rules
✔ I can find HCF and LCM
✔ I understand word problems
✔ I can explain answers clearly
Ultra-Final Summary
Numbers follow patterns
Factors divide, multiples repeat
Prime numbers are building blocks
HCF and LCM solve real-life problems
Strong basics lead to exam success
FINAL END NOTE
Mastering Playing with Numbers makes mathematics logical, easy, and enjoyable. This chapter builds a strong foundation for Class 7 ICSE Mathematics and future learning.
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Finding the Smallest Number Divisible by Given Numbers
Such questions are based on LCM.
Example
Find the smallest number divisible by 4, 6 and 9.
4 = 2²
6 = 2 × 3
9 = 3²
LCM = 2² × 3² = 36
Finding the Greatest Number That Divides Given Numbers
Such questions are based on HCF.
Example
Find the greatest number that divides 48 and 64 exactly.
48 = 2⁴ × 3
64 = 2⁶
HCF = 2⁴ = 16
Equal Remainder Questions (ICSE Favourite)
When a number leaves the same remainder, subtract the numbers and find HCF.
Example
Find the greatest number that divides 35 and 50, leaving the same remainder.
50 − 35 = 15
HCF of 15 = 15
Why Co-Prime Numbers Are Important
Their HCF is always 1
They simplify fractions
Used in algebra and number theory
Example
8 and 15 are co-prime
HCF = 1
Properties of Co-Prime Numbers
They may be odd–odd, even–odd, or odd–even
They need not be prime themselves
Example
14 and 25 are co-prime
More Challenging Divisibility Checks
Example
Is 2,376 divisible by 3 and 4?
Sum of digits = 2 + 3 + 7 + 6 = 18 → divisible by 3
Last two digits = 76 → divisible by 4
✔ The number is divisible by 12
Combined Divisibility Rule
A number divisible by 3 and 4 is divisible by 12.
Example
312 → divisible by 12 ✔
Concept of Lowest Factor and Highest Multiple
Lowest factor of any number = 1
Highest multiple of a number = Not defined (infinite)
Mathematical Reasoning Questions
Q
Why is 1 neither prime nor composite?
Answer:
Because it has only one factor, not two or more.
Concept-Based MCQs
Which number has exactly three factors?
a) 6
b) 8
c) 9
d) 10
Answer: c) 9
HCF of two co-prime numbers is
a) 0
b) 1
c) The greater number
d) The smaller number
Answer: b) 1
Fill in the Blanks
The smallest prime number is _ → 2 The HCF of two prime numbers is → 1
Multiples of a number are __ → Infinite
True or False
Every prime number is odd ❌
Every composite number has more than two factors ✔
LCM of two numbers is always greater than both ❌
Match the Following
Column A
Column B
Prime number
Two factors
Composite number
More than two factors
HCF
Greatest factor
LCM
Smallest multiple
Five-Mark ICSE Question (Model Answer)
Question
Find the HCF and LCM of 18 and 24.
Solution:
18 = 2 × 3²
24 = 2³ × 3
HCF = 2 × 3 = 6
LCM = 2³ × 3² = 72
How This Chapter Helps in Higher Classes
Used in fractions
Needed for algebra
Foundation for number systems
Helps in problem solving
Teacher’s Exam Tip
Always:
Show working
Use correct method
Box the final answer
Write units if needed
Quick Memory Tricks
HCF → Divide
LCM → Repeat
Prime → Two factors
Composite → Many factors
One-Page Rapid Revision
Factors divide
Multiples repeat
Prime numbers are special
Co-prime → HCF = 1
HCF × LCM = Product
Final Student Checklist
✔ Can I identify prime numbers?
✔ Can I find HCF & LCM by all methods?
✔ Can I solve word problems?
✔ Can I apply divisibility rules fast?
SUPER FINAL CONCLUSION
The chapter Playing with Numbers is the backbone of Class 7 ICSE Mathematics. A clear understanding of this chapter improves accuracy, speed, and confidence. Mastery of these concepts ensures success not only in school exams but also in higher mathematical learning.
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Word Problems Based on HCF
These problems test logical thinking and real-life application.
Example
Three ropes are 24 m, 36 m, and 48 m long. Find the maximum length of rope that can be cut from each without wasting any length.
Solution:
HCF of 24, 36, and 48
24 = 2³ × 3
36 = 2² × 3²
48 = 2⁴ × 3
HCF = 2² × 3 = 12 m
Word Problems Based on LCM
LCM is used when events repeat together.
Example
Two bells ring every 6 seconds and 8 seconds. When will they ring together again?
LCM of 6 and 8
= 24
✔ Bells will ring together after 24 seconds
Difference Between HCF and LCM (Exam Question)
HCF
LCM
Greatest common factor
Least common multiple
Smaller value
Larger value
Used for dividing
Used for repeating
Relationship Formula (Very Important)
For two numbers:
HCF × LCM = Product of the numbers
Example
Numbers: 12 and 18
HCF = 6
LCM = 36
6 × 36 = 216
12 × 18 = 216 ✔
Finding a Number When HCF and LCM Are Given
Example
HCF = 5, LCM = 60
If one number is 15, find the other number.
Product = 5 × 60 = 300
Other number = 300 ÷ 15 = 20
Higher Order Thinking Skills (HOTS)
Question
Can two numbers have the same HCF and LCM?
Answer:
Yes, when both numbers are equal.
Example:
12 and 12
HCF = 12, LCM = 12
Reasoning Questions
Q
Why is the LCM of two co-prime numbers equal to their product?
Answer:
Because co-prime numbers have no common factors except 1.
Activity-Based Learning
Activity 1
Write all factors of numbers from 1 to 20 and classify them as:
Prime
Composite
Neither
Activity 2
List the first 10 multiples of 6 and 9 and circle the common multiples.
Mental Maths Practice
HCF of 9 and 27 = _ LCM of 4 and 10 =
Smallest prime number = __
Answers:
9 2) 20 3) 2
Case Study–Based Question
A school arranges students in rows of 24 or 36 without any student left out.
Question
What is the maximum number of students in each row?
Solution:
HCF of 24 and 36 = 12
Very Short Answer Questions
Write the smallest composite number.
→ 4
Write the HCF of two prime numbers.
→ 1
Is 91 a prime number?
→ No (7 × 13)
Short Answer Questions
Find the LCM of 5, 10, and 20.
→ 20
Write two co-prime numbers.
→ 8 and 15
Long Answer Question
Question
Find the HCF and LCM of 45 and 75.
45 = 3² × 5
75 = 3 × 5²
HCF = 3 × 5 = 15
LCM = 3² × 5² = 225
Common Mistakes Students Make
❌ Confusing factors with multiples
❌ Forgetting prime factorization
❌ Wrong application of divisibility rules
❌ Not checking final answer
Examiner’s Tip
ICSE examiners give step marks.
Always:
Write factorization clearly
Use correct symbols
Draw tables neatly
One-Day Before Exam Revision
✔ Revise divisibility rules
✔ Practice HCF–LCM sums
✔ Learn definitions by heart
✔ Solve 5 word problems
Viva Voce Questions
What is a factor?
What is the smallest prime number?
Define co-prime numbers.
State the HCF–LCM relationship.
Fun Maths Fact
Every prime number greater than 3 can be written in the form 6n ± 1.
Practice Worksheet (Homework)
Find the HCF of 30 and 45.
Find the LCM of 6 and 14.
Find the smallest number divisible by 8, 12, and 16.
Two numbers have HCF = 4 and LCM = 48. Find the numbers.
Answer Key
15
42
48
12 and 16
Student Confidence Builder
If you understand:
Factors ✔
Multiples ✔
Prime numbers ✔
HCF & LCM ✔
Then 90% of this chapter is mastered 🎯
FINAL FINAL WRAP-UP
Playing with Numbers develops:
Logical thinking
Calculation speed
Problem-solving ability
It is one of the most scoring chapters in Class 7 ICSE Mathematics if practiced well.
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🧪 Case-study based questions
🏆 Olympiad-level problems
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Here is EVEN MORE EXTENDED CONTENT, continuing seamlessly and helping you reach 3000+ words easily. This section adds exam-ready worksheets, Olympiad touch, competency-based questions, and self-evaluation tools. You can append it directly after the previous part.

Competency-Based Questions (As per ICSE Pattern)
Question 1
A number is divisible by 2, 3, and 5. Which of the following must be true?
It ends with 0
The sum of digits is divisible by 3
✔ Correct Answer: Both statements are true
Question 2
The HCF of two numbers is 1. What can you say about the numbers?
Answer:
The numbers are co-prime.
Assertion–Reason Questions
Q
Assertion (A): 2 is the only even prime number.
Reason (R): All even numbers except 2 are divisible by 2.
✔ Answer:
Both A and R are true, and R correctly explains A.
Fill in the Blanks
The smallest whole number is _ The HCF of two co-prime numbers is
A number divisible by 10 always ends with __
Answers:
0 2) 1 3) 0
Match the Following
Column A
Column B
Prime number
Has exactly two factors
Composite number
Has more than two factors
Co-prime numbers
HCF is 1
Even number
Divisible by 2
True or False
Every multiple of a number is greater than the number. → True
1 is a composite number. → False
LCM is always greater than HCF. → True
Olympiad Corner (Easy Level)
Question
Find the smallest number which when divided by 12, 15, and 20 leaves remainder 0.
Solution:
LCM of 12, 15, 20
= 60
Number Puzzles
Puzzle
I am a two-digit number.
I am divisible by 2 and 3.
The sum of my digits is 6.
✔ Answer: 24
Real-Life Application Question
A shopkeeper packs biscuits in packets of 6, 9, or 12 without leftovers.
Question
What is the minimum number of biscuits?
Answer:
LCM of 6, 9, and 12 = 36
Table-Based Question
Complete the table:
Number
Prime / Composite
2
Prime
9
Composite
17
Prime
21
Composite
Self-Evaluation Checklist
Tick ✔ if you can:
Define prime numbers
Apply divisibility rules
Find HCF using factorisation
Solve word problems on LCM
✔✔✔✔ = Chapter Mastered
Mistake Alert Box (Very Important)
⚠ Never confuse:
HCF ≠ LCM
Factors ≠ Multiples
Prime ≠ Co-prime
Short Trick for Exams
👉 If numbers are co-prime
Then:
HCF = 1
LCM = Product of numbers
Revision Table (One-Page View)
Concept
Key Point
Natural Numbers
Start from 1
Whole Numbers
Include 0
Prime Numbers
Exactly 2 factors
HCF
Greatest common factor
LCM
Least common multiple
Practice Set – Level 1
Find HCF of 16 and 24
Find LCM of 3 and 7
Is 51 divisible by 3?
Answers:
8 2) 21 3) Yes
Practice Set – Level 2
Find the smallest number divisible by 5, 10, and 20
Find two co-prime numbers between 10 and 20
Answers:
20
11 and 12
Think and Answer
Why is 1 neither prime nor composite?
Answer:
Because it has only one factor.
Teacher’s Observation
Students who practice divisibility rules daily solve HCF–LCM problems 50% faster.
Homework Assignment
✔ Write divisibility rules from 2 to 10
✔ Solve 10 HCF problems
✔ Solve 10 LCM problems
Last-Minute Exam Strategy
⏰ 1 day before exam:
Revise formulas
Practice 5 word problems
Read definitions once
GRAND CONCLUSION
The chapter Playing with Numbers is the foundation of arithmetic.
A strong grip on this chapter makes fractions, algebra, and number systems easy in higher classes.
If you master: ✔ Factors
✔ Multiples
✔ Divisibility
✔ HCF & LCM
👉 Maths becomes fun and scoring! 🌟
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I can now prepare:
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Exam-Oriented Model Questions (ICSE Style)
Question 1 (2 Marks)
Define a composite number. Give two examples.
Model Answer:
A composite number is a number which has more than two factors.
Examples: 4, 6
Question 2 (3 Marks)
State the divisibility rules for 3 and 9.
Model Answer:
A number is divisible by 3 if the sum of its digits is divisible by 3.
A number is divisible by 9 if the sum of its digits is divisible by 9.
Question 3 (4 Marks)
Find the HCF of 20, 30, and 40 using prime factorisation.
Solution:
20 = 2² × 5
30 = 2 × 3 × 5
40 = 2³ × 5
HCF = 2 × 5 = 10
Step-by-Step Method Writing (Very Important)
While solving HCF/LCM:
Write the given numbers clearly
Show prime factorisation
Highlight common factors
Write the final answer with unit
✔ This ensures full step marks.
Competency Mapping (NEP-Aligned)
Skill
How Chapter Helps
Logical Thinking
Divisibility rules
Problem Solving
Word problems
Numerical Fluency
Factors & multiples
Real-Life Application
HCF & LCM
Value-Based Question
Three students distribute 24 chocolates equally.
Question
What value does this situation show?
Answer:
It shows sharing and equality, which can be solved using HCF.
Reflection Question (For Students)
After studying this chapter, answer:
Can I find HCF confidently?
Can I apply divisibility rules quickly?
Can I explain prime numbers in my own words?
✔ If yes, you are on the right track.
Maths Lab Activity (ICSE Suggested)
Activity
Take number cards from 1 to 50.
Separate prime numbers
Circle multiples of 5
Highlight even numbers
Learning Outcome:
Students understand number classification visually.
Case Study – Advanced
A factory produces items in batches of 18, 24, and 36.
Question
What is the largest batch size possible so that all batches are equal?
Solution:
HCF of 18, 24, and 36 = 6
Brain Booster Question
Which is greater:
LCM of (6, 8) OR LCM of (8, 12)?
LCM (6, 8) = 24
LCM (8, 12) = 24
✔ Both are equal
Error Analysis Question
A student says 1 is a prime number.
Correct the Statement
Incorrect, because a prime number must have exactly two factors.
Very Important Definitions (Revise Before Exam)
Factor: A number that divides another number completely
Multiple: Result of multiplying a number by natural numbers
HCF: Greatest common factor
LCM: Least common multiple
Speed Maths Section
Without calculation:
Is 245 divisible by 5? → Yes
Is 312 divisible by 3? → Yes
Is 420 divisible by 10? → Yes
Comparative Question
Why is LCM always greater than or equal to the given numbers?
Answer:
Because it is a common multiple, formed by multiplying numbers.
Integrated Learning
This chapter helps in:
Fractions (simplification)
Algebra (factorisation)
Mensuration (equal division)
Data handling (grouping)
Worksheet – Mixed Practice
Find HCF of 14 and 28
Find LCM of 9 and 12
Write all factors of 18
Is 121 a prime number?
Answers:
14
36
1, 2, 3, 6, 9, 18
No (11 × 11)
Challenge Yourself
Find the smallest number divisible by 7, 14, and 21.
Answer:
LCM = 42
Concept Strength Test
If:
You can solve HCF in one method
You can solve LCM in two methods Then your concept is strong 💪
Teacher’s Rubric (How You Are Assessed)
Criteria
Marks
Concept clarity
✔✔
Method
✔✔
Accuracy
✔✔
Presentation
✔
Learning Outcomes
After completing this chapter, students can: ✔ Classify numbers
✔ Apply divisibility rules
✔ Solve real-life problems
✔ Build strong arithmetic base
Mind Map (Textual)
Playing with Numbers →
Factors → Prime/Composite → Divisibility → HCF → LCM → Applications
FINAL EXTENDED SUMMARY
The chapter Playing with Numbers is not just about calculations.
It develops:
Accuracy
Speed
Reasoning
Confidence
Mastering this chapter makes Class 7 ICSE Mathematics easier and more scoring.
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Class 7 mathematics playing with Numbers easy notes

Class 7 ICSE Mathematics: Playing with Numbers – Complete Study Guide

H2: Introduction to Playing with Numbers