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):**