Number Play
“Number Play” explores the patterns and properties hidden inside whole numbers. This Class 8 chapter revisits factors and multiples, prime and composite numbers, handy divisibility rules, and the ideas of HCF and LCM, using puzzles and patterns to build a feel for how numbers behave.
Learning objectives
- Find factors and multiples of numbers.
- Identify prime and composite numbers.
- Apply divisibility rules for 2, 3, 4, 5, 9 and 10.
- Find the HCF and LCM of numbers.
Key concepts
Factors and multiples
A factor of a number divides it exactly with no remainder, while a multiple is got by multiplying the number by a whole number. So 1, 2, 3 and 6 are factors of 6, and 6, 12, 18 are multiples of 6. Every number is a factor of itself and a multiple of 1.
Prime and composite numbers
A prime number has exactly two factors, 1 and itself (2, 3, 5, 7, 11…), while a composite number has more than two factors (4, 6, 8, 9…). The number 1 is neither prime nor composite, and 2 is the only even prime. Any composite number can be written as a product of primes.
Divisibility rules
Divisibility rules let us test factors quickly: a number is divisible by 2 if it ends in an even digit; by 5 if it ends in 0 or 5; by 10 if it ends in 0; by 3 or 9 if the sum of its digits is divisible by 3 or 9; and by 4 if its last two digits form a number divisible by 4.
HCF and LCM
The Highest Common Factor (HCF) is the largest number that divides two or more numbers, and the Lowest Common Multiple (LCM) is the smallest number that is a multiple of all of them. Both can be found using prime factorisation, and for any two numbers, HCF × LCM = product of the numbers.
Important formulas
Divisible by 3 / 9
if the digit sum is divisible by 3 / 9
Divisible by 4
if the last two digits form a multiple of 4
HCF–LCM relation
HCF × LCM = product of the two numbers
Key definitions
- Factor
- A number that divides another exactly, leaving no remainder.
- Multiple
- A number obtained by multiplying a given number by a whole number.
- Prime number
- A number with exactly two factors: 1 and itself.
- HCF
- The Highest Common Factor — the largest number dividing two or more numbers.
Solved examples
Q1. Is 4128 divisible by 4?
Solution: The last two digits form 28, and 28 ÷ 4 = 7, so yes, 4128 is divisible by 4.
Q2. Find the HCF of 12 and 18.
Solution: 12 = 2×2×3, 18 = 2×3×3. Common factors 2 and 3 give HCF = 2×3 = 6.
Q3. Is 91 prime or composite?
Solution: 91 = 7 × 13, so it has more than two factors; it is composite.
Common mistakes to avoid
- Calling 1 a prime number — it is neither prime nor composite.
- Confusing factors (divide the number) with multiples (the number divides them).
- Forgetting 2 is prime because it is even.
- Mixing up HCF (largest divisor) with LCM (smallest common multiple).
Number Play — MCQ Quiz
10 questions with instant feedback. Use number keys 1–4 to answer.
How many factors does a prime number have?
Practice questions
Short answer
State the divisibility rule for 9.
A number is divisible by 9 if the sum of its digits is divisible by 9.
What is the difference between a factor and a multiple?
A factor divides the number exactly; a multiple is obtained by multiplying the number by a whole number.
Write the prime factorisation of 60.
60 = 2 × 2 × 3 × 5.
Long answer
Explain how to find the HCF and LCM of 12 and 18 by prime factorisation.
Write each as primes: 12 = 2×2×3 and 18 = 2×3×3. For the HCF, take the common prime factors with the lowest powers: one 2 and one 3, giving HCF = 6. For the LCM, take all prime factors with the highest powers: 2×2 from 12 and 3×3 from 18, giving LCM = 2×2×3×3 = 36. Check: HCF × LCM = 6 × 36 = 216 = 12 × 18.
List the divisibility rules for 2, 3, 4, 5, 9 and 10 with an example each.
Divisible by 2 if it ends in an even digit (e.g. 48). By 5 if it ends in 0 or 5 (e.g. 65). By 10 if it ends in 0 (e.g. 120). By 3 if the digit sum is divisible by 3 (e.g. 51, since 5+1=6). By 9 if the digit sum is divisible by 9 (e.g. 153, since 1+5+3=9). By 4 if the last two digits form a multiple of 4 (e.g. 316, since 16 is divisible by 4).
HOTS (Higher Order Thinking)
A number is divisible by both 2 and 3. What other number must divide it, and why?
It must be divisible by 6, because 6 = 2 × 3 and the number contains both factors.
Why does every composite number have at least one prime factor?
A composite number can be broken into smaller factors repeatedly; this process stops at primes, so the original number is built from prime factors.
Quick revision
Revision notes
- Factor divides exactly; multiple is the number times a whole number.
- Prime: two factors; composite: more than two; 1 is neither.
- Divisibility: 2 (even), 5 (0/5), 10 (0), 3 & 9 (digit sum), 4 (last two digits).
- HCF = largest common factor, LCM = smallest common multiple; HCF × LCM = product.
Key takeaways
- Divisibility rules speed up finding factors.
- Prime factorisation underlies HCF and LCM.
- 2 is the only even prime; 1 is neither prime nor composite.
Frequently asked questions
Is 1 a prime number?
No — it has only one factor, so it is neither prime nor composite.
How do I quickly test divisibility by 3?
Add the digits; if the sum is divisible by 3, so is the number.
What is the relation between HCF and LCM?
For two numbers, HCF × LCM equals the product of the numbers.