LCM: Formulas, Methods, Tricks and Solved Examples

LCM, or least common multiple, is the smallest positive number exactly divisible by two or more given numbers. This page explains prime factorisation and division methods, the HCF-LCM relationship, fraction rules, useful shortcuts and solved LCM questions for quantitative aptitude.

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What is LCM?

The LCM of two or more positive integers is the smallest positive number that is exactly divisible by each of them.

For example, the multiples of 6 are 6, 12, 18, 24, 30, 36, and the multiples of 8 are 8, 16, 24, 32, 40. Therefore, LCM(6, 8) = 24. Using prime factorisation, take every prime factor with its greatest exponent appearing in any given number.

LCM Formula & Tricks

Important Formulas

Prime factorisation formula
LCM = product of each prime factor raised to its greatest exponent

For 12 = 2² × 3 and 18 = 2 × 3², LCM = 2² × 3² = 36.

HCF-LCM relationship
LCM(a, b) × HCF(a, b) = a × b

For two positive integers, the product of their LCM and HCF equals the product of the numbers.

LCM of two coprime numbers
LCM(a, b) = a × b, when HCF(a, b) = 1

For example, 8 and 15 are coprime, so their LCM is 8 × 15 = 120.

LCM of fractions in lowest terms
LCM(a/b, c/d) = LCM(a, c) / HCF(b, d)

This rule applies when the fractions are first reduced to their lowest terms. For 2/3 and 5/6, LCM = LCM(2, 5) / HCF(3, 6) = 10/3.

Quick Tricks

Use the product-HCF shortcut

For two numbers, divide their product by their HCF to obtain the LCM. This avoids complete prime factorisation when the HCF is easy to find.

Example: LCM(24, 36) = (24 × 36) / HCF(24, 36) = 864 / 12 = 72.
Multiply only the required prime factors

In prime factorisation, do not multiply repeated factors more than their greatest required exponent.

Example: For 18 = 2 × 3² and 24 = 2³ × 3, use 2³ × 3² = 72, not the product of all listed factors.
Use the product for coprime numbers

If two numbers have no common factor other than 1, their LCM is simply their product.

Example: HCF(14, 25) = 1, so LCM(14, 25) = 14 × 25 = 350.

LCM Concepts

LCM by Prime Factorisation

Factor each number into primes and multiply every prime factor using its greatest exponent.

Write the numbers as products of prime factors. Select each distinct prime appearing in the factorizations, take its highest power, and multiply these powers. For 20 = 2² × 5 and 30 = 2 × 3 × 5, LCM = 2² × 3 × 5 = 60.

Example: LCM(16, 24) = LCM(2⁴, 2³ × 3) = 2⁴ × 3 = 48.

LCM by the Division Method

Divide all the given numbers by a common prime factor repeatedly until every remaining number becomes 1; the product of the divisors is the LCM.

A divisor may divide one or more current numbers. Numbers not divisible by that divisor are brought down unchanged. The product of all divisors used gives the least common multiple.

Example: For 12 and 18: divide by 2 to get 6 and 9, divide by 2 to get 3 and 9, divide by 3 to get 1 and 3, then divide by 3 to get 1 and 1. Thus LCM = 2 × 2 × 3 × 3 = 36.

Relation Between LCM and HCF

For two positive integers, LCM = (product of the numbers) ÷ HCF.

The formula is LCM(a, b) = ab/HCF(a, b). It is valid for two numbers and gives the LCM directly after calculating their HCF.

Example: For 45 and 60, HCF = 15. Hence, LCM = (45 × 60)/15 = 180.

LCM of Fractions

For fractions in lowest terms, divide the LCM of the numerators by the HCF of the denominators.

First reduce every fraction to its lowest form. Then use LCM of numerators divided by HCF of denominators. For 3/4 and 5/6, LCM = LCM(3, 5)/HCF(4, 6) = 15/2.

Example: LCM(4/9, 10/21) = LCM(4, 10)/HCF(9, 21) = 20/3, after confirming both fractions are in lowest terms.

LCM Video Lessons

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LCM Using Prime Factorisation and Ladder Method

Learn to find the least common multiple using prime factorisation and the ladder method, with clear steps for applying both techniques to number system problems.

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Practice LCM Questions

Practise published questions related to this topic.

1The number nearest to 43582 divisible by each of 25, 50 and 75 is -→ 2Two numbers are such that their HCF is 18 and their LCM is 540. If one of the numbers is 90, find the other number.→ 358P8 is divisible by 9. What is the value of P?→ 4A rectangular courtyard, 4.56 meters long and 7.68 meters wide is paved exactly with square tiles of same size. Find the size (in cm) of the largest tile that can be used for this purpose.→ 5Least Common Multiple of A and B is 20. Least Common Multiple of C and D is 30. What is the Least Common Multiple of A, B, C and D?→ 6If √(18.49) + √(29 + x2) = 19.3, then what is the value of x?→ 7Two bells in a school ring at 8 am. One bell rings after every 30 minutes and the other after every 40 minutes. If the school gets over at 2 pm, how many times in a day do the bells ring together?→ 8What is the least number which should be added to 539, so that the number obtained is exactly divisible by 10, 15 and 3?→ 9(√0.0169/1.3) × (0.18/√3.24) × (0.24/√0.0576) = ?→ 10What is the Highest Common Factor of 0.48, 0.64 and 0.80?→

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Quick Revision Notes

LCM Quick Revision

Use these rules to calculate the least common multiple accurately.

  • LCM is the smallest positive common multiple of the given numbers.
  • In prime factorisation, select every prime with its greatest exponent.
  • For two positive integers, LCM × HCF = product of the integers.
  • If two numbers are coprime, their LCM equals their product.
  • In the division method, multiply all divisors used until every quotient becomes 1.
  • For fractions in lowest terms, LCM = LCM of numerators ÷ HCF of denominators.
  • The LCM of a number and one is the number itself: LCM(n, 1) = n.

LCM FAQs

What is the LCM of 12 and 15?

12 = 2² × 3 and 15 = 3 × 5. Therefore, LCM(12, 15) = 2² × 3 × 5 = 60.

How is LCM calculated using HCF?

For two positive integers, use LCM(a, b) = (a × b) ÷ HCF(a, b). For 18 and 30, LCM = (18 × 30) ÷ 6 = 90.

What is the LCM of 8, 12 and 20?

8 = 2³, 12 = 2² × 3 and 20 = 2² × 5. Taking the greatest powers gives LCM = 2³ × 3 × 5 = 120.

What is the difference between LCM and HCF?

LCM is the smallest positive number divisible by all the given numbers. HCF is the greatest positive number that divides all the given numbers exactly.

What is the LCM of two coprime numbers?

It is their product because their HCF is 1. Thus, LCM(9, 16) = 9 × 16 = 144.

How do you find the LCM of fractions?

After reducing the fractions to lowest terms, use LCM of numerators divided by HCF of denominators. Thus, LCM(2/3, 4/5) = LCM(2, 4) ÷ HCF(3, 5) = 4.

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