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AFCAT LCM and HCF

Study Material — 6 PYQs (2018–2023) · Concept Notes · Shortcuts

AFCAT LCM and HCF is a frequently tested subtopic — 6 previous year questions from 2018–2023 papers are included below with concept notes, key rules and shortcut tricks.

6 PYQs
2018–2023
34 Practice
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10 Key Points
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Previous Year Questions

AFCAT LCM and HCF — Past Exam Questions

6 questions from actual AFCAT papers · all shown free · click option to reveal solution

Exam Q 12018Previous Year Pattern

Find the LCM of 12, 18, and 24.

Exam Q 22018Previous Year Pattern

The HCF of two numbers is 12 and their LCM is 360. If one of the numbers is 72, what is the other number?

Exam Q 32018Previous Year Pattern

The LCM of two numbers is 2520 and their HCF is 12. If one number is 168, find the other number.

Exam Q 42023Previous Year Pattern

The LCM of two numbers is 12 times their HCF. If the sum of the HCF and LCM is 403, find the HCF of the two numbers.

Exam Q 52023Previous Year Pattern

The HCF of two numbers is 16 and their LCM is 480. How many pairs of numbers satisfy this condition?

Exam Q 62023Previous Year Pattern

Three bells ring at intervals of 18 minutes, 24 minutes, 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?

Concept Notes

LCM and HCF— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic

CORE CONCEPT LCM and HCF are two of the most tested topics in SSC CGL. Understanding them well means free marks every year.

HCF (Highest Common Factor): The largest number that divides two or more numbers exactly. Also called GCD (Greatest Common Divisor). LCM (Least Common Multiple): The smallest number that is exactly divisible by two or more numbers.

Simple Memory Trick: HCF is always SMALLER than or equal to the given numbers. LCM is always LARGER than or equal to the given numbers. ---

2.Key Rules

Core rules you must know cold
Rule 1

HCF x LCM = Product of the two numbers. This only works for TWO numbers, not three or more

Rule 2

HCF of fractions = HCF of numerators / LCM of denominators Rule 3: LCM of fractions = LCM of numerators / HCF of denominators Rule 4: HCF always divides LCM completely. LCM is always a multiple of HCF

Rule 5

For consecutive integers, HCF is always 1. ---

3.Formula Block

Memorise — at least one formula appears in every paper
HCF x LCM = First Number x Second Number
HCF of fractions = HCF of numerators / LCM of denominators
LCM of fractions = LCM of numerators / HCF of denominators
Answer = LCM of (a, b, c) + r

Largest number that divides a, b, c leaving remainders r1, r2, r3:

Find HCF of (a - r1), (b - r2), (c - r3)

Largest number that divides a, b, c leaving SAME remainder:

Find HCF of (a - b), (b - c), (a - c)

Smallest number divisible by a, b, c leaving remainder r:

---

4.Exam Patterns

What examiners ask — read before attempting PYQs
Example

Divides 17, 23, 35 leaving same remainder. Differences = 6, 12, 18. HCF = 6

Trick 2 — LCM Shortcut for Bells

Bells ring at intervals of a, b, c minutes. They ring together after LCM(a, b, c) minutes

Trick 3 — Missing Number

If HCF = H and LCM = L, and one number = N, then other number = (H x L) / N ---

5.Worked Example

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

Use formula — HCF x LCM = Product of numbers

2
Step 2

12 x LCM = 36 x 48

3
Step 3

12 x LCM = 1728

4
Step 4

LCM = 1728 / 12 = 144 Answer: LCM = 144 --- WORKED EXAMPLE 2 Question: Find the largest number that divides 245, 1029, and 1417 leaving remainder 5 each time.

1
Step 1

Subtract remainder from each number. 245 - 5 = 240 1029 - 5 = 1024 1417 - 5 = 1412

2
Step 2

Find HCF of 240, 1024, 1412. 240 = 2 x 2 x 2 x 2 x 3 x 5 1024 = 2^10 1412 = 2 x 2 x 353

3
Step 3

Common factor = 2 x 2 = 4 Answer: The largest number is 4. ---

6.Exam Traps

Common mistakes students make — avoid these

— THE NUMBER 1 TRAP Students forget that HCF x LCM = Product of numbers works ONLY for TWO numbers. They apply it to three numbers and get the wrong answer. If three numbers are given, you must find HCF and LCM separately using prime factorisation.

Never use the product formula for three or more numbers. This single mistake causes lakhs of students to lose marks every year.

Key Points to Remember

  • HCF is the largest number that divides given numbers exactly without leaving a remainder.
  • LCM is the smallest number that is exactly divisible by all the given numbers.
  • Formula: HCF x LCM = Product of two numbers (valid ONLY for two numbers).
  • HCF of fractions = HCF of numerators divided by LCM of denominators.
  • LCM of fractions = LCM of numerators divided by HCF of denominators.
  • Largest number dividing a, b, c with SAME remainder = HCF of (a-b), (b-c), (a-c).
  • Smallest number divisible by a, b, c leaving remainder r = LCM(a, b, c) + r.
  • HCF always divides LCM completely — LCM is always a multiple of HCF.
  • HCF of any two consecutive integers is always 1.
  • Missing number when HCF, LCM and one number are known = (HCF x LCM) divided by the known number.

Exam-Specific Tips

  • HCF x LCM = Product of the two numbers — this formula is valid ONLY for exactly two numbers, not three or more.
  • HCF of fractions formula: HCF of numerators / LCM of denominators.
  • LCM of fractions formula: LCM of numerators / HCF of denominators.
  • If two numbers are co-prime (HCF = 1), their LCM equals their product.
  • The HCF of any set of numbers is always a factor of their LCM.
  • Bells ringing together again after time T: T = LCM of all individual ringing intervals.
  • Largest number dividing a, b, c leaving different remainders r1, r2, r3 = HCF of (a-r1), (b-r2), (c-r3).
  • For prime numbers p and q, HCF is always 1 and LCM is always p x q.
Practice MCQs

LCM and HCF — Practice Questions

34graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 34

All MCQs →
Practice 1easy

Three numbers are in the ratio 2:3:4. If their HCF is 5, find the largest number.

Practice 2easy

The HCF of two numbers is 8 and their LCM is 96. How many pairs of numbers satisfy this condition?

Practice 3easy

Two bells ring at intervals of 18 minutes and 24 minutes respectively. If they ring together at 9:00 AM, at what time will they ring together again?

Practice 4easy

The HCF of two numbers is 15. If the numbers are 45 and 75, verify the HCF is correct. What is their LCM?

Practice 5easy

The HCF of two numbers is 15. If the numbers are 45 and 75, verify which statement is correct.

Practice 6easy

Two bells ring at intervals of 18 minutes and 24 minutes respectively. If they ring together at 9:00 AM, at what time will they ring together again?

Practice 7easy

The HCF of two numbers is 18. If the numbers are in the ratio 2:3, find the larger number.

Practice 8easy

Find the LCM of 24, 36, and 48.

Practice 9easy

Find the LCM of 24, 36, and 40.

Practice 10easy

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

Practice 11easy

The HCF of two numbers is 8. Which of the following cannot be their LCM?

Practice 12easy

Find the HCF of 56, 84, and 140 using the prime factorization method.

Practice 13medium

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?

Practice 14medium

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?

Practice 15medium

The HCF of two numbers is 16 and the sum of the two numbers is 144. If one number is 48, what is the other number?

Practice 16medium

Two numbers have an HCF of 18 and an LCM of 540. How many such pairs of numbers exist?

Practice 17medium

The LCM of two numbers is 168 and their HCF is 12. If the difference between the two numbers is 60, find the larger number.

Practice 18medium

Three numbers are in the ratio 2:3:4. Their LCM is 120. What is the sum of the three numbers?

Practice 19medium

Three bells ring at intervals of 18 minutes, 24 minutes, and 36 minutes respectively. If they all ring together at 9:00 AM, at what time will they ring together again?

Practice 20medium

Two numbers have an HCF of 15 and an LCM of 180. How many pairs of such numbers are possible?

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60-Second Revision — LCM and HCF

  • Formula: HCF x LCM = Product of TWO numbers only — never use this for three or more numbers.
  • Fractions — Remember: HCF goes numerator/denominator(LCM), LCM goes numerator(LCM)/denominator(HCF).
  • Trap: Applying HCF x LCM = Product formula to three numbers is the most common and costly mistake — avoid it.
  • Remainder trick: Same remainder problem — find HCF of differences of the given numbers.
  • Bells/Timing problems — always find LCM of the time intervals given.
  • Remember: HCF is always less than or equal to the smallest number; LCM is always greater than or equal to the largest number.
  • Co-prime numbers have HCF = 1, so their LCM = their product directly.
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