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

Worked Examples

### Example 1: Find HCF and LCM of 18 and 24

**Prime factorisation:**

18 = 2 × 3²
24 = 2³ × 3

**HCF** = Common primes with lowest powers = 2¹ × 3¹ = 6

**LCM** = All primes with highest powers = 2³ × 3² = 8 × 9 = 72

**Verification:** HCF × LCM = 6 × 72 = 432 = 18 × 24 ✓

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### Example 2: Three bells ring at intervals of 4, 6, and 9 minutes. If they ring together at 9:00 AM, when will they ring together again?

**Solution:** Find LCM of 4, 6, and 9.

4 = 2²
6 = 2 × 3
9 = 3²

LCM = 2² × 3² = 4 × 9 = 36 minutes

**Answer:** They will ring together at 9:36 AM.

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### Example 3: Find HCF of 2/3, 4/5, and 6/7.

**Formula:** HCF of fractions = HCF of numerators ÷ LCM of denominators

HCF of 2, 4, 6 = 2
LCM of 3, 5, 7 = 105

**Answer:** HCF = 2/105

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### Example 4: The HCF of two numbers is 12 and their LCM is 180. If one number is 36, find the other.

**Using formula:** HCF × LCM = Product of numbers

12 × 180 = 36 × other number
2160 = 36 × other number
Other number = 2160 ÷ 36 = 60

**Answer:** 60

Common Mistakes

| Wrong Thinking | Correct Approach | |----------------|------------------| | Taking highest powers for HCF | HCF uses **lowest** powers of **common** primes only | | Taking lowest powers for LCM | LCM uses **highest** powers of **all** primes present | | Applying product formula to three numbers | HCF × LCM = a × b works only for **two** numbers | | Confusing HCF/LCM of fractions | Remember: HCF uses HCF on top, LCM on bottom; LCM is opposite | | Forgetting to verify answer | Always check: HCF must divide both numbers; LCM must be divisible by both | | In word problems, using HCF when LCM is needed | "Together again" or "at the same time" → LCM; "Largest piece" or "maximum distribution" → HCF |

Quick Reference

**HCF = common primes, lowest powers; LCM = all primes, highest powers**

**HCF × LCM = Product of two numbers** (most important shortcut)

**"Bells ring together again" = Find LCM**

**"Largest tile for floor" or "Maximum equal distribution" = Find HCF**

**HCF of fractions: HCF(num) ÷ LCM(den); LCM of fractions: LCM(num) ÷ HCF(den)**

**HCF ≤ both numbers ≤ LCM** (always true — use for elimination)

**Co-primes: HCF = 1, so LCM = product**

LCM and HCF