SSC Notes / Quantitative Aptitude / Chapter 3
Chapter 3 of 27

HCF & LCM

Factorisation and division methods for HCF and LCM, the product formula, and classic word-problem patterns.

📖 ~12 min read 🔢 SSC Quantitative Aptitude

Introduction

HCF (Highest Common Factor) and LCM (Lowest Common Multiple) questions appear both directly and hidden inside word problems (bell-ringing, tank-filling, and traffic-light-style questions) — a compact but high-utility SSC Quant chapter.

Key Formula

Flowchart — HCF-LCM Product Relationship
HCF(a,b) × LCM(a,b) = a × b

This formula only holds for exactly two numbers — not for three or more.

Methods to Find HCF

MethodProcess
Factorisation MethodExpress each number as a product of primes; HCF = product of common prime factors (lowest powers)
Division Method (Euclid's Algorithm)Divide larger number by smaller; then divide previous divisor by the remainder; repeat until remainder = 0. Last divisor = HCF
Q. Find the HCF of 84 and 126 using the division method.
126 ÷ 84 → remainder 42
84 ÷ 42 → remainder 0
HCF = 42

Methods to Find LCM

MethodProcess
Factorisation MethodLCM = product of all prime factors (highest powers appearing in any number)
Formula Method (2 numbers)LCM = (a × b) ÷ HCF(a,b)
Q. Find the LCM of 84 and 126 (using HCF = 42 from above).
LCM = (84 × 126) ÷ 42 = 10584 ÷ 42 = 252

HCF/LCM of Fractions

Formula
HCF of fractions = HCF(numerators) ÷ LCM(denominators)
LCM of fractions = LCM(numerators) ÷ HCF(denominators)

Classic Word Problem Pattern — "Bells Ring Together"

Q. Three bells ring at intervals of 6, 8, and 12 minutes. If they ring together at 9:00 AM, when will they next ring together?
Find LCM(6, 8, 12) = 24
They will ring together again after 24 minutes → at 9:24 AM

Pattern rule: "Ring/beep/change together again" questions always use LCM.

Classic Word Problem Pattern — "Maximum Size/Largest Group"

Q. Find the greatest length that can measure 48 m, 60 m, and 72 m exactly.
Find HCF(48, 60, 72) = 12
Greatest measuring length = 12 m

Pattern rule: "Greatest/maximum size that divides exactly" questions always use HCF.
💡 Exam Tip: Quickly identify the question type — "together again," "same time," "least number" → LCM; "greatest," "maximum," "largest measure/group" → HCF. This pattern recognition saves crucial seconds.
✅ Practice Focus: HCF×LCM = product formula (2 numbers only) · Euclid's division method for HCF · LCM/HCF of fractions · Word-problem pattern recognition (bells → LCM, greatest measure → HCF).

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