LCM and HCF — Study Notes for Bihar TET Paper I

Overview

LCM (Lowest Common Multiple) and HCF (Highest Common Factor) form the backbone of number-system problems in Bihar TET Paper I Mathematics. These concepts test a candidate's understanding of divisibility, factors, and multiples — skills essential for teaching primary-level arithmetic.

In the exam, expect 2–3 direct questions on finding LCM/HCF using various methods, plus application-based word problems involving time intervals, distribution of items, or measurement conversions. Mastery here also supports related topics like fractions, ratio-proportion, and simplification.

For classroom teaching, these concepts help children understand why numbers behave the way they do when grouped, shared, or repeated — making this topic both exam-critical and pedagogically significant.

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 products obtained by multiplying a number by natural numbers. Example: 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 numbers.

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

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

• **HCF of fractions** = HCF of numerators ÷ LCM of denominators.

• **LCM of fractions** = LCM of numerators ÷ HCF of denominators.

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, LCM = product of the numbers | | HCF of consecutive numbers | Always 1 | | LCM of consecutive numbers | Product of the numbers |

**Three methods to find HCF:** 1. Prime factorisation — Take common prime factors with lowest powers 2. Division method — Divide larger by smaller, then divisor by remainder, repeat until remainder is 0 3. Listing factors — List all factors and pick the highest common one

**Two methods to find LCM:** 1. Prime factorisation — Take all prime factors with highest powers 2. Division method — Divide by primes, continue until all quotients become 1