LCM and HCF

Overview

LCM (Lowest Common Multiple) and HCF (Highest Common Factor) form the backbone of number theory questions in HP TET Mathematics. These concepts appear directly in 2–4 questions and also underpin problems on fractions, ratio-proportion, and word problems involving time, bells ringing together, or distributing items equally.

Mastery requires two things: fluency with the methods (prime factorisation, division method, relationship formula) and the ability to recognise which concept a word problem demands. LCM questions typically involve "when will events coincide?" scenarios, while HCF questions involve "largest possible equal parts" situations.

For HP TET, focus on numbers up to 3–4 digits. Speed matters—practice until you can find LCM and HCF of two numbers in under 30 seconds.

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Key Concepts

• **Factor**: A number that divides another exactly. Factors of 12: 1, 2, 3, 4, 6, 12.

• **Multiple**: The product of a number and any whole number. Multiples of 4: 4, 8, 12, 16, ...

• **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.

• **Co-prime numbers**: Two numbers whose HCF is 1 (e.g., 8 and 15). Their LCM equals their product.

• **Fundamental relationship**: For any two numbers a and b: **HCF × LCM = a × b**

• **HCF of given numbers ≤ smallest number**; **LCM of given numbers ≥ largest number**.

• **HCF divides LCM**: The HCF of any set of numbers always divides their LCM exactly.

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Formulas / Key Facts

| Concept | Formula / Fact | |---------|----------------| | Product relationship | HCF(a, b) × LCM(a, b) = a × b | | Finding LCM when HCF known | LCM = (a × b) ÷ HCF | | Finding HCF when LCM known | HCF = (a × b) ÷ LCM | | HCF of fractions | HCF of numerators ÷ LCM of denominators | | LCM of fractions | LCM of numerators ÷ HCF of denominators | | Co-prime numbers | HCF = 1, so LCM = a × b | | HCF by division | Divide larger by smaller; divide divisor by remainder; repeat until remainder = 0 | | Prime factorisation for HCF | Take lowest power of common primes | | Prime factorisation for LCM | Take highest power of all primes |

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Worked Examples

### Example 1: Find HCF and LCM of 36 and 48

**Prime Factorisation Method**

36 = 2² × 3² 48 = 2⁴ × 3¹