LCM and HCF

Overview

LCM (Lowest Common Multiple) and HCF (Highest Common Factor) form a fundamental arithmetic topic that appears consistently in JKTET Paper I Mathematics. This topic tests your understanding of divisibility, factors, multiples, and the relationship between numbers—skills essential for a primary-level mathematics teacher.

For the JKTET exam, you need to master three things: finding LCM and HCF using different methods, understanding the relationship between LCM and HCF, and applying these concepts to word problems involving time, distance, and measurement. Questions typically range from direct calculation to application-based problems involving real-life scenarios relevant to the J&K context (like calculating intervals for buses, dividing resources equally among students, etc.).

The pedagogy section of JKTET also expects you to know how to teach these concepts to primary students using concrete materials and local examples, making conceptual clarity doubly important.

Key Concepts

• **Factors** are numbers that divide a given number exactly (without remainder). For 12, factors are 1, 2, 3, 4, 6, 12.

• **Multiples** are numbers obtained by multiplying a given number by natural numbers. Multiples of 4 are 4, 8, 12, 16, 20...

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

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

• **Co-prime numbers** have HCF = 1 (example: 8 and 15). Their LCM equals their product.

• **The product relationship**: For any two numbers a and b, LCM × HCF = a × b. This is a frequently tested formula.

• **HCF of given numbers is always less than or equal to the smallest number**; LCM is always greater than or equal to the largest number.

• **Prime factorisation method** works for both LCM and HCF: HCF uses common prime factors with lowest powers; LCM uses all prime factors with highest powers.

Formulas / Key Facts

**Core Formula:** LCM(a, b) × HCF(a, b) = a × b

**Finding HCF — Three Methods:**

1. **Listing factors method**: List all factors of each number, identify common factors, pick the highest.

2. **Prime factorisation method**: Express each number as product of primes. HCF = product of common prime factors with lowest powers.

3. **Division method (Euclid's algorithm)**: Divide larger by smaller, then divisor by remainder, repeat until remainder is 0. Last divisor is HCF.