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REET L1+L2 · Maths

LCM (Lowest Common Multiple) and HCF (Highest Common Factor)

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This page brings together every formula, method and shortcut for LCM (Lowest Common Multiple) and HCF (Highest Common Factor) you need for REET. The Number System section brings a direct calculation or word problem from this topic almost every year.

  • 1 question in REET 2024
  • Class 4–6 level
  • 7 min read
Verified 17 September 2026

How REET asks this topic

REET 2024 carried a direct calculation question from this topic — finding the ratio of the LCM and HCF of two numbers. Going by the pattern, expect one direct calculation question (based on prime factorisation or the division method) and occasionally a word problem (bells ringing together, distributing objects equally). The formula LCM × HCF = product of the two numbers is the single most-tested point.

Definition — HCF and LCM

The Highest Common Factor (HCF, also called GCD) of two or more numbers is the largest factor that divides every one of those numbers exactly.

The Lowest Common Multiple (LCM) of two or more numbers is the smallest multiple that is exactly divisible by every one of those numbers. HCF is always less than or equal to the given numbers, while LCM is always greater than or equal to them.

Methods for Finding HCF and LCM

MethodUse for HCFUse for LCM
Prime Factorisation MethodProduct of the common prime factors raised to their lowest powersProduct of all prime factors (common and uncommon) raised to their highest powers
Division Method (Euclid's Method)Divide the larger number by the smaller, then keep dividing by the remainder; the last divisor is the HCFDivide the numbers successively by common prime factors; the product of all divisors and the final quotients gives the LCM

Applying this to 16 and 36: 16 = 2⁴, 36 = 2² × 3² ⇒ HCF = 2² = 4, LCM = 2⁴ × 3² = 144

Formulas Worth Remembering

For two numbers
LCM × HCF = first number × second number (this rule holds only for two numbers, not for three or more)
Co-prime numbers
HCF is always 1; LCM = product of both numbers
HCF of fractions
HCF of numerators ÷ LCM of denominators
LCM of fractions
LCM of numerators ÷ HCF of denominators

Recognising HCF or LCM in Word Problems

When a question uses words like 'equal pieces', 'maximum number to distribute into', or 'largest possible measure' — that's an HCF situation. For example: finding the largest possible length to cut several ropes of different lengths into equal pieces.

When a question uses words like 'happening together', 'minimum number', or 'meeting/colliding again' — that's an LCM situation. For example: several bells that ring at different intervals — when will they all ring together again — this is found using LCM.

Some questions also involve 'with a remainder' word problems — where dividing a number by several other numbers leaves the same fixed remainder each time. Here, you first find the LCM and then add the remainder (for the smallest such number), or subtract the remainder from the given dividends before finding the HCF (when every dividend leaves the same remainder).

Solved Examples

Example 1: Find the HCF and LCM of 24 and 36. 24 = 2³ × 3, 36 = 2² × 3². HCF = product of common prime factors at their lowest power = 2² × 3 = 12. LCM = product of all prime factors at their highest power = 2³ × 3² = 72. Check: HCF × LCM = 12 × 72 = 864 = 24 × 36 ✓.

Example 2: Three bells ring at intervals of 9, 12 and 15 minutes respectively. If all three ring together at 8:00 am, when will they next ring together? Solution: 9 = 3², 12 = 2² × 3, 15 = 3 × 5. LCM = 2² × 3² × 5 = 180 minutes = 3 hours. So all three bells will next ring together at 11:00 am.

Common Mistakes and Things to Watch For

The most common mistake is miscounting a power while doing prime factorisation — for example, stopping at 36 = 2×18 without breaking 18 further into 18=2×9=2×3×3. Always keep factorising until every factor is prime.

The second common mistake is, while using the division method for HCF, taking the last non-zero remainder as the answer, when the correct answer is actually the last divisor at which the remainder becomes zero. In word problems too, don't rush past the signal words for 'sharing equally/largest measure' (HCF) versus 'happening together/repeating' (LCM).

How to Teach This to Children

Classroom activity

Give children two groups of pebbles or counters and have them form equal-sized piles — the largest pile that works gives a hands-on feel for HCF. For LCM, repeatedly join two coloured strips of different lengths end to end and mark where both strips' ends first coincide — that point is the LCM.

Teaching aids

A factor-tree chart, a number line with multiples marked on strips, and coloured counting beads are all useful here.

Common misconception

Students often mix up which direction HCF and LCM go, or wrongly apply the rule 'LCM × HCF = product' to three numbers as well — this rule is only correct for two numbers.

Assessment idea

Give a real-life situation (like a school bell, or laying floor tiles) and ask 'do we need HCF or LCM here, and why?' — this checks conceptual understanding.

Memory trick / exam tip

If you know the HCF of two numbers and their product, find the LCM directly as LCM = product ÷ HCF — no need to repeat the whole process. But remember, this shortcut only works for two numbers; for three or more numbers, always solve step by step.

REET 2024 Questions (with Solutions)

Question 107

The ratio of least common multiple and highest common factor of 16 and 36 is

  1. A36 : 1
  2. B16 : 1
  3. C1 : 36
  4. D72 : 1

Explanation

16 = 2⁴ और 36 = 2² × 3²। म.स. (HCF) = 2² = 4 तथा ल.स. (LCM) = 2⁴ × 3² = 144। अतः LCM : HCF = 144 : 4 = 36 : 1। विकल्प (C) 1 : 36 उलटा अनुपात (HCF : LCM) है।

Practice Questions

Question 1

What is the HCF of 18 and 24?

  1. A6
  2. B12
  3. C72
  4. D3

Explanation

18 = 2 × 3², 24 = 2³ × 3. The common prime factors at their lowest power = 2¹ × 3¹ = 6. So HCF = 6.

Question 2

Which is the most suitable initial teaching method to clarify the concept of HCF and LCM in primary classes?

  1. Aमूर्त वस्तुओं (जैसे कंकड़, गोटियाँ) से समान समूह बनवाना व गुणनखंड वृक्ष बनवाना
  2. Bकेवल सूत्र रटवाना
  3. Cसीधे बड़ी संख्याओं के प्रश्न हल कराना
  4. Dकेवल परिभाषा बोर्ड पर लिखवाना

Explanation

Using concrete objects (like pebbles or counters) to form equal groups, and building factor trees, gives children a solid, hands-on experience of HCF and LCM — something rote formulas alone can't provide. This follows the concrete-to-abstract teaching principle.

Question 3

The LCM of two co-prime numbers is 91. If one number is 7, what is the other number?

  1. A13
  2. B7
  3. C91
  4. D84

Explanation

The HCF of co-prime numbers is always 1, so LCM = product of both numbers. 91 = 7 × 13, so the other number is 13.

Question 4

The HCF of two numbers is 4 and LCM is 48. If one number is 16, what is the other number?

  1. A12
  2. B16
  3. C8
  4. D24

Explanation

LCM × HCF = first number × second number ⇒ 48 × 4 = 16 × second number ⇒ second number = 192 ÷ 16 = 12.

Question 5

What is the HCF of 60, 75 and 90?

  1. A15
  2. B5
  3. C30
  4. D45

Explanation

60 = 2² × 3 × 5, 75 = 3 × 5², 90 = 2 × 3² × 5. The lowest common power across all three = 3 × 5 = 15.

Question 6

What is the LCM of 12, 15 and 20?

  1. A60
  2. B120
  3. C30
  4. D180

Explanation

12 = 2² × 3, 15 = 3 × 5, 20 = 2² × 5. The highest power of all prime factors = 2² × 3 × 5 = 60.

Question 7

Three bells toll at intervals of 9, 12 and 15 minutes. If they toll together at 8:00 am, when will they toll together again?

  1. A11:00 बजे
  2. B10:00 बजे
  3. C9:30 बजे
  4. D12:00 बजे

Explanation

LCM(9,12,15) = 180 minutes = 3 hours (9=3², 12=2²×3, 15=3×5 ⇒ LCM=2²×3²×5=180). So 8:00 + 3:00 = 11:00 am is when all three will ring together again.

Question 8

Find the greatest number which divides 245 and 1029 leaving remainder 5 in each case.

  1. A16
  2. B15
  3. C32
  4. D8

Explanation

You need the HCF of 245−5=240 and 1029−5=1024. 240 = 2⁴×3×5, 1024 = 2¹⁰. The lowest common power = 2⁴ = 16.

Question 9

A teacher wants to distribute 36 notebooks and 60 pencils equally among students with nothing left over. What is the maximum number of students?

  1. A12
  2. B18
  3. C6
  4. D24

Explanation

This is a question of HCF(36,60). 36 = 2²×3², 60 = 2²×3×5. HCF = 2²×3 = 12. So the maximum number of students it can be divided among is 12.

Question 10

What is the HCF of 2/3 and 4/9?

  1. A2/9
  2. B4/3
  3. C2/3
  4. D8/27

Explanation

HCF of fractions = HCF of numerators ÷ LCM of denominators = HCF(2,4) ÷ LCM(3,9) = 2 ÷ 9 = 2/9.

Frequently Asked Questions

How many questions come from LCM-HCF in REET?+

Usually 1 question from this topic appears in the Number System section — as a direct calculation or a word problem; REET 2024 also had 1 question.

What's the main difference between HCF and LCM?+

HCF is the largest common factor of the given numbers, while LCM is their smallest common multiple. HCF is always less than or equal to the numbers, and LCM is always greater than or equal to them.

What does the product of two numbers' HCF and LCM equal?+

It equals the product of those two numbers themselves — this rule does not apply to three or more numbers.

What is the HCF of co-prime numbers?+

The HCF of co-prime numbers is always 1, even if the numbers themselves aren't prime — for example, 8 and 9 are co-prime, though neither is a prime number.

How do you find the LCM of fractions?+

LCM of fractions = LCM of the numerators ÷ HCF of the denominators.

How do you find HCF using the division method?+

Divide the larger number by the smaller one, then keep dividing the divisor by the remainder — repeat this until the remainder becomes zero; the last divisor is the HCF.

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Change Log

  • 17 September 2026 — English page published — mirrors the fact-checked Hindi page, independently bilingual-accuracy checked