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SSC CPO LCM and HCF

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.

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2019–2019
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Previous Year Questions

SSC CPO LCM and HCF — Past Exam Questions

7 questions from actual SSC CPO papers · all shown free · click option to reveal solution

Exam Q 12019Previous Year Pattern

The HCF of two numbers is 18. If the numbers are 54 and 90, verify the HCF and find their LCM.

Exam Q 22019Previous Year Pattern

What is the LCM of 24, 36, and 48?

Exam Q 32019Previous Year Pattern

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

1.Core Concept

Read this first — the foundation of the topic

LCM (Least Common Multiple) and HCF (Highest Common Factor) are fundamental concepts in Number System that appear in almost every SSC CGL paper. Understanding these concepts is crucial for solving time and work, ratio, and simplification problems. 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.

2.Key Rules

Core rules you must know cold
1

For two numbers a and b: HCF × LCM = a × b

2

HCF is always less than or equal to the smallest number

3

LCM is always greater than or equal to the largest number

4

If two numbers are co-prime (HCF = 1), then LCM = product of numbers

3.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
LCM = (a × b) ÷ HCF(a,b)

HCF by Division Method: Keep dividing larger by smaller until remainder is zero

4.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.

5.Worked Example

Solve this step-by-step before moving on
1
Step 1

Prime factorization 48 = 2⁴ × 3¹ 72 = 2³ × 3²

2
Step 2

HCF = product of lowest powers = 2³ × 3¹ = 8 × 3 = 24

3
Step 3

LCM = product of highest powers = 2⁴ × 3² = 16 × 9 = 144 Verification: HCF × LCM = 24 × 144 = 3456 = 48 × 72 ✓ Worked Example 2: Two bells ring at intervals of 15 and 20 minutes. If they ring together at 9:00 AM, when will they ring together next?

1
Step 1

Find LCM of 15 and 20 15 = 3 × 5 20 = 2² × 5

2
Step 2

LCM = 2² × 3 × 5 = 60 minutes

3
Step 3

Next time = 9:00 AM + 60 minutes = 10:00 AM

6.Shortcuts

Use these to save 30–60 seconds per question
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).

7.Exam Traps

Common mistakes students make — avoid these

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
  • Prime factorization method: HCF uses lowest powers, LCM uses highest powers
  • Division method for HCF: divide larger by smaller repeatedly until remainder is zero
  • Word problems: LCM for 'together again', HCF for 'equal groups'
  • Three number formula: HCF × LCM ≠ a × b × c (common exam trap)
  • Speed check: if numbers share no common factors, they are co-prime

Exam-Specific Tips

  • HCF of two consecutive numbers is always 1
  • HCF of two consecutive even numbers is always 2
  • LCM of fractions = LCM of numerators ÷ HCF of denominators
  • HCF of fractions = HCF of numerators ÷ LCM of denominators
  • For three numbers a, b, c: HCF(a,b,c) divides LCM(a,b,c)
  • Product of two numbers = HCF × LCM (valid only for exactly two numbers)
  • HCF is also called GCD (Greatest Common Divisor)
  • LCM is also called LCM (Least Common Multiple) or SCM (Smallest Common Multiple)

60-Second Revision — LCM and HCF

  • Formula: HCF × LCM = a × b (only for two numbers, not three)
  • Remember: Prime factorization uses lowest powers for HCF, highest for LCM
  • Trap: Three number product rule does NOT apply
  • Shortcut: Co-prime numbers have HCF = 1, LCM = their product
  • Pattern: LCM problems = 'when together again', HCF = 'maximum equal parts'
  • Quick check: HCF ≤ smaller number, LCM ≥ larger number
  • Speed tip: Use division method for HCF of two numbers
Divisibility Rules→
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SSC CPO LCM and HCF — Study Material, 7 PYQs & Practice MCQs | ZestExam