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CBSE • Class 6 • Mathematics • Ch 3

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Playing with Numbers

In CBSE Class 6 Mathematics, "Playing with Numbers" delves into number theory. Students explore factors and multiples, prime and composite numbers, the Sieve of Eratosthenes, divisibility tests (for 2, 3, 4, 5, 6, 8, 9, 10, 11), prime factorization, Highest Common Factor (HCF / GCD), Lowest Common Multiple (LCM), and real-world word problems using HCF and LCM.

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Overview Core Concepts Formulas & Identities Solved Examples Common Traps Summary Self Check Take MCQ Test

🧩 Have You Ever Wondered?

How can you tell if the 8-digit number 9,876,543 is divisible by 9 in just 3 seconds without actually dividing it?

Divisibility rules and prime numbers are the foundational secrets of modern encryption, banking cryptography, and computer security! Explore the magical universe of factors and multiples.

Why This Chapter Matters

Before You Begin (Prerequisites)

Multiplication tables up to 20
Basic division and remainder concept

What You Will Learn (Core Objectives)

Define and list Factors and Multiples of any given number.
Distinguish Prime numbers from Composite numbers; identify Twin Primes and Co-prime numbers.
Apply Divisibility Rules for 2, 3, 4, 5, 6, 8, 9, 10, and 11.
Perform Prime Factorization using factor trees and division methods.
Compute HCF (GCD) and LCM using prime factorization and common division.
Solve real-world word problems (traffic lights, room tiles, milk containers) using HCF and LCM.

Chapter Roadmap & Progression

1 1. Factors, Multiples, Prime & Comp...

2 2. Rapid Divisibility Rules

3 3. HCF, LCM & Real-World Word Probl...

Complete Concept Guide (100% Curriculum Coverage)

1. Factors, Multiples, Prime & Composite Numbers

Factor: An exact divisor of a number that leaves zero remainder. $1$ is a factor of every number; every number is a factor of itself; factors are finite in number.
Multiple: A number obtained by multiplying given number by an integer. Multiples are infinite.
Prime Numbers: Numbers greater than 1 with exactly two distinct factors: 1 and the number itself ($2, 3, 5, 7, 11, 13, 17, 19 \dots$). $2$ is the smallest prime and the only even prime number!
Composite Numbers: Numbers having more than two factors ($4, 6, 8, 9, 10, 12 \dots$). $1$ is neither prime nor composite!
Co-prime Numbers: Two numbers having only 1 as a common factor (e.g. 4 and 9).
Twin Primes: Two prime numbers differing by 2 (e.g. 3 and 5, 5 and 7, 11 and 13).

2. Rapid Divisibility Rules

Divisor	Divisibility Rule	Example
2	Units digit is even: 0, 2, 4, 6, or 8.	$4,896$ is divisible (ends in 6).
3	Sum of all digits is a multiple of 3.	$7,341 \to 7+3+4+1=15$ (divisible by 3).
4	Number formed by last two digits is divisible by 4.	$1,936 \to 36 \div 4 = 9$ (divisible).
5	Units digit is either 0 or 5.	$8,975$ is divisible.
6	Divisible by BOTH 2 and 3 simultaneously.	$246$ (even $\to$ div by 2; $2+4+6=12 \to$ div by 3).
8	Number formed by last three digits is divisible by 8.	$5,128 \to 128 \div 8 = 16$ (divisible).
9	Sum of all digits is a multiple of 9.	$5,472 \to 5+4+7+2 = 18$ (divisible).
10	Units digit is strictly 0.	$9,430$ is divisible.
11	Difference between sum of digits at odd places and sum at even places is either 0 or a multiple of 11.	$1,331 \to (1+3) - (3+1) = 4 - 4 = 0$ (divisible).

3. HCF, LCM & Real-World Word Problems

HCF (Highest Common Factor / GCD): The largest factor common to two or more numbers. Found by multiplying lowest powers of common prime factors.

LCM (Lowest Common Multiple): The smallest positive number that is a multiple of all given numbers. Found by multiplying highest powers of all prime factors.

Golden Formula for Two Numbers:$$\text{Product of Two Numbers } (a \times b) = \text{HCF}(a, b) \times \text{LCM}(a, b)$$

Word Problem Decision Strategy

Use HCF when dividing/grouping items into maximum equal sizes without remainder (e.g., maximum capacity of container, largest tile size).
Use LCM when finding the next simultaneous event or smallest common point (e.g., traffic lights blinking together, runners meeting at starting line).

Key Formulas, Identities & Theorems

HCF-LCM Product Law

$$a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b)$$

Applies strictly to any pair of positive integers.

Divisibility by 11

$$|\sum d_{\text{odd}} - \sum d_{\text{even}}| \in \{0, 11, 22, \dots\}$$

Alternating place difference condition.

Conceptual Solved Examples & Case Studies

Example 1

Three traffic lights at different road crossings change after every 48 seconds, 72 seconds, and 108 seconds respectively. If they change simultaneously at 7:00 a.m., at what time will they change simultaneously again?

Step-by-Step Solution:

The time interval for simultaneous change is the LCM of 48, 72, and 108.
Prime factorization:
$48 = 2^4 \times 3$
$72 = 2^3 \times 3^2$
$108 = 2^2 \times 3^3$
$\text{LCM} = 2^4 \times 3^3 = 16 \times 27 = 432\text{ seconds}$.
Converting to minutes: $432\text{ seconds} = 7\text{ minutes } 12\text{ seconds}$.
Therefore, they change simultaneously again at $\mathbf{7:07:12\text{ a.m.}}$.

Example 2

Find the greatest number which divides 615 and 963 leaving a remainder of 6 in each case.

Step-by-Step Solution:

Subtract the remainder 6 from both numbers:
$615 - 6 = 609$
$963 - 6 = 957$
Now find $\text{HCF}(609, 957)$.
$609 = 3 \times 7 \times 29$
$957 = 3 \times 11 \times 29$
Common prime factors are 3 and 29.
$\text{HCF} = 3 \times 29 = \mathbf{87}$.

Common Misconceptions & Examiner Traps

Common Misconception

Classifying 1 as a prime number.

Scientific Reality & Correction

A prime number MUST have exactly TWO distinct factors (1 and itself). 1 has only one factor (itself), so 1 is NEITHER prime NOR composite.

Common Misconception

Using HCF when asked about bells ringing together.

Scientific Reality & Correction

Bells ring at repeating periodic intervals; to find when they coincide next, you must compute the LCM, not HCF.

Visual Learning & Conceptual Map

HCF vs LCM Strategic Matrix

Factors (Division/Maximums) vs Multiples (Repetition/Minimums)

HCF (Highest Common Factor)

• Product of smallest powers of common prime factors.
• Used for "maximum capacity", "largest size", "dividing equally".

LCM (Lowest Common Multiple)

• Product of highest powers of all prime factors.
• Used for "ringing together", "smallest number", "repeating cycles".

Chapter Summary & 10 Key Takeaways

Takeaway 1

Factors divide a number completely; multiples are products with integers.

Takeaway 2

Primes have exactly 2 factors; composites have more than 2. 2 is the only even prime.

Takeaway 3

Divisibility rules test 2, 3, 4, 5, 6, 8, 9, 10, and 11 quickly.

Takeaway 4

HCF is the highest common factor; LCM is the lowest common multiple.

Takeaway 5

Product of two numbers equals HCF × LCM.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

Check if 1,082,418 is divisible by 11.

Reveal Answer & Explanation

Answer: Sum of digits at odd places (from right): $8 + 4 + 8 + 1 = 21$. Sum of digits at even places: $1 + 2 + 0 = 3$. Difference $= 21 - 3 = 18$. Since 18 is not divisible by 11, the number is NOT divisible by 11.
Check difference of odd and even digit sums.

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