Study Material — 7 PYQs (2019–2019) · Concept Notes · Shortcuts
SSC CPO LCM and HCF is a frequently tested subtopic — 7 previous year questions from 2019–2019 papers are included below with concept notes, key rules and shortcut tricks.
Three bells ring at intervals of 18 minutes, 24 minutes, and 30 minutes respectively. If they all ring together at 9:00 AM, at what time will they ring together again?
Exam Q 42019Previous Year Pattern
Two numbers have an HCF of 8 and an LCM of 168. How many pairs of such numbers exist?
Exam Q 52019Previous Year Pattern
The HCF of two numbers is 15. If the numbers are in the ratio 3:5, what is their LCM?
Exam Q 62019Previous Year Pattern
Three bells ring at intervals of 18, 24, and 36 minutes respectively. If they all ring together at 9:00 AM, at what time will they ring together again for the third time?
Exam Q 72019Previous Year Pattern
Two numbers are in the ratio 5:7. Their LCM is 245. What is the HCF of these two numbers?
Concept Notes
LCM and HCF— Rules & Concept
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Core Concept
Read this first — the foundation of the topic
→Core Concept
HCF is the largest number that divides all given numbers completely. LCM is the smallest number that is divisible by all given numbers. Think of HCF as the biggest common piece, and LCM as the smallest common whole
💡Key Properties
• For two numbers a and b: HCF × LCM = a × b
• HCF is always less than or equal to the smallest number
• LCM is always greater than or equal to the largest number
• If two numbers are co-prime (HCF = 1), then LCM = product of numbers
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Formula Block
Memorise — at least one formula appears in every paper
HCF × LCM = Product of two numbers
For three numbers: HCF(a,b,c) × LCM(a,b,c) ≠ a × b × c
HCF by Division Method: Keep dividing larger by smaller until remainder is zero
LCM = (a × b) ÷ HCF(a,b)
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Exam Patterns
What examiners ask — read before attempting PYQs
📋SSC CGL typically asks
direct HCF/LCM calculation, word problems involving bells/traffic lights, finding numbers when HCF/LCM is given, and ratio-based problems. Most questions are 2-mark difficulty level
⚡Powerful Shortcut - Product Rule
For any two numbers, if you know any three values among {number1, number2, HCF, LCM}, you can find the fourth instantly using: HCF × LCM = number1 × number2
Next time = 9:00 AM + 60 minutes = 10:00 AM
Time-Saving Trick - Division Method:
For HCF: Keep dividing larger number by smaller until remainder is zero. Last divisor is HCF.
48 ÷ 72: 72 = 48 × 1 + 24, then 48 = 24 × 2 + 0. HCF = 24
⚡Another Speed Trick - Co-prime Check
If two numbers have no common factors except 1, their HCF = 1 and LCM = their product
✏️Examples
(7,11), (15,16), (25,28)
⚠️#1 Most Common Mistake
Students confuse the formula application for three numbers. The golden rule HCF × LCM = product applies ONLY to two numbers. For three or more numbers, this formula does NOT work. Always work with pairs or use prime factorization method for multiple numbers.
This mistake costs students 2-4 marks per exam
💡Practical Tip
In word problems, LCM gives 'together again' scenarios (bells, traffic lights). HCF gives 'maximum equal groups' scenarios (distribution problems). Identify the scenario type first, then apply the appropriate concept.
Key Points to Remember
HCF × LCM = Product of two numbers (works only for exactly two numbers)
HCF is the largest common divisor, LCM is the smallest common multiple
For co-prime numbers: HCF = 1, LCM = product of the numbers
LCM = (a × b) ÷ HCF for any two numbers a and b
HCF ≤ smallest number, LCM ≥ largest number among given numbers