LCM and HCF

Overview

LCM (Lowest Common Multiple) and HCF (Highest Common Factor) form the backbone of number-system questions in CG TET Paper I Mathematics. These concepts test a candidate's understanding of divisibility, factors, and multiples—fundamental skills required to teach primary-level arithmetic.

In CG TET, expect 2–3 direct questions on LCM and HCF, often presented as word problems involving time intervals, distribution of items, or measurement scenarios. Mastery here also supports related topics like fractions, ratio-proportion, and simplification. The pedagogy angle requires you to understand how children conceptualize factors and multiples through concrete examples before moving to abstract methods.

Success demands knowing multiple methods (prime factorization, division, listing), recognizing when to apply LCM versus HCF, and avoiding common calculation errors under exam pressure.

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Key Concepts

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

• **Multiples** are numbers obtained by multiplying a given number by 1, 2, 3, ... Example: Multiples of 4 are 4, 8, 12, 16, ...

• **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 numbers.

• **Co-prime numbers** have HCF = 1. Example: 8 and 15 are co-prime.

• **Fundamental relationship**: For any two numbers a and b, HCF(a, b) × LCM(a, b) = a × b. This formula is a frequent exam shortcut.

• **HCF ≤ both numbers ≤ LCM** always. HCF divides both numbers; both numbers divide the LCM.

• **HCF of co-primes is 1; LCM of co-primes is their product.**

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Formulas / Key Facts

| Concept | Formula / Fact | |---------|----------------| | Product Rule | HCF × LCM = Product of the two numbers | | Finding LCM | LCM = Product ÷ HCF | | Finding HCF | HCF = Product ÷ LCM | | HCF of fractions | HCF of numerators ÷ LCM of denominators | | LCM of fractions | LCM of numerators ÷ HCF of denominators | | HCF by division | Divide larger by smaller; continue with remainder until remainder = 0; last divisor is HCF | | LCM by prime factorization | Take highest power of each prime factor | | HCF by prime factorization | Take lowest power of common prime factors |

**Quick fact**: If one number is a factor of the other, HCF = smaller number, LCM = larger number.