Factors and Multiples

Overview

Factors and multiples form the backbone of number theory at the primary level and appear consistently in WB TET Paper I Mathematics. This topic tests whether candidates understand divisibility relationships and can apply them to solve problems involving HCF (Highest Common Factor) and LCM (Lowest Common Multiple).

For the WB TET, you must be comfortable classifying numbers as prime or composite, finding all factors of a given number, listing multiples, and computing HCF and LCM using multiple methods. Questions often combine these concepts with word problems involving real-life situations like distributing items equally or finding common time intervals.

Mastery here also builds the foundation for teaching fractions, simplification, and ratio-proportion at the primary level—skills directly tested in the pedagogy section as well.

Key Concepts

• **Factor**: A number that divides another number exactly (without remainder). Example: 4 is a factor of 12 because 12 ÷ 4 = 3 exactly.

• **Multiple**: A number obtained by multiplying a given number by any whole number. Example: Multiples of 5 are 5, 10, 15, 20, ...

• **Prime Number**: A number greater than 1 that has exactly two factors—1 and itself. Examples: 2, 3, 5, 7, 11, 13.

• **Composite Number**: A number greater than 1 that has more than two factors. Examples: 4, 6, 8, 9, 12.

• **Special case**: 1 is neither prime nor composite. 2 is the only even prime number.

• **Co-prime (Relatively Prime)**: Two numbers whose HCF is 1. Example: 8 and 15 are co-prime.

• **HCF (Highest Common Factor)**: The largest number that divides two or more numbers exactly. Also called GCD (Greatest Common Divisor).

• **LCM (Lowest Common Multiple)**: The smallest number that is a multiple of two or more numbers.

Formulas / Key Facts

**Product relationship for two numbers:** HCF × LCM = Product of the two numbers (For numbers a and b: HCF(a,b) × LCM(a,b) = a × b)

**Finding HCF — Prime Factorisation Method:** Write each number as a product of primes → Take common primes with lowest powers → Multiply them.

**Finding LCM — Prime Factorisation Method:** Write each number as a product of primes → Take all primes with highest powers → Multiply them.

**Finding HCF — Division Method (Euclidean Algorithm):** Divide larger by smaller → Divide divisor by remainder → Repeat until remainder is 0 → Last divisor is HCF.

**Divisibility Rules (essential for quick factorisation):**

• By 2: Last digit is 0, 2, 4, 6, or 8
• By 3: Sum of digits divisible by 3
• By 4: Last two digits divisible by 4
• By 5: Last digit is 0 or 5
• By 6: Divisible by both 2 and 3
• By 9: Sum of digits divisible by 9
• By 11: Difference of sums of alternate digits is 0 or divisible by 11

**Number of factors formula:** If n = p^a × q^b × r^c, then total factors = (a+1)(b+1)(c+1)

Worked Examples

**Example 1: Find HCF and LCM of 24 and 36 using prime factorisation.**

Step 1 — Prime factorise each number: 24 = 2 × 2 × 2 × 3 = 2³ × 3¹ 36 = 2 × 2 × 3 × 3 = 2² × 3²

Step 2 — HCF: Take common primes with lowest powers. Common primes: 2 and 3 HCF = 2² × 3¹ = 4 × 3 = 12

Step 3 — LCM: Take all primes with highest powers. LCM = 2³ × 3² = 8 × 9 = 72

Verification: HCF × LCM = 12 × 72 = 864; Product = 24 × 36 = 864 ✓

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**Example 2: Find HCF of 56 and 98 using division method.**

Step 1: Divide 98 by 56 98 = 56 × 1 + 42 (remainder = 42)

Step 2: Divide 56 by 42 56 = 42 × 1 + 14 (remainder = 14)

Step 3: Divide 42 by 14 42 = 14 × 3 + 0 (remainder = 0)

HCF = 14 (last divisor before remainder becomes 0)

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**Example 3: Word Problem** Three bells ring at intervals of 6, 9, and 12 minutes. If they ring together at 8:00 AM, when will they ring together again?

Solution: Find LCM of 6, 9, and 12. 6 = 2 × 3 9 = 3² 12 = 2² × 3

LCM = 2² × 3² = 4 × 9 = 36 minutes

They will ring together again at 8:36 AM.

Common Mistakes

• **Confusing factors and multiples** → Remember: Factors divide into (are smaller or equal), multiples are products of (are larger or equal). "Factors are few, multiples are many."

• **Forgetting that 1 is neither prime nor composite** → Students often call 1 a prime number. It has only one factor (itself), not two, so it does not qualify as prime.

• **Swapping HCF and LCM methods** → HCF uses lowest powers of common primes; LCM uses highest powers of all primes. Mixing these gives wrong answers.

• **Applying the product rule to more than two numbers** → HCF × LCM = Product works only for two numbers. For three or more numbers, this shortcut fails.

• **Skipping verification** → Always verify using the product relationship (for two numbers) or by checking divisibility. This catches calculation errors quickly.

• **Missing the "0" as a multiple** → Technically, 0 is a multiple of every number. But in school mathematics, we usually list positive multiples starting from the number itself.

Quick Reference

• Factor divides exactly; multiple is the product.
• Prime = exactly 2 factors; Composite = more than 2 factors; 1 = neither.
• HCF × LCM = Product of the two numbers.
• HCF: common primes, lowest powers. LCM: all primes, highest powers.
• 2 is the smallest and only even prime number.
• Co-prime numbers have HCF = 1.