HCF and LCM: Formulas, Methods, Tricks and Questions

HCF and LCM are used to find common factors and common multiples of numbers. This topic covers prime factorisation, the division method, standard formulas, HCF-LCM relationships and applications in problems involving repeated events, grouping and fractions. Worked examples show how to calculate both values accurately and select the correct method.

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What are HCF and LCM?

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

For example, the factors of 12 are 1, 2, 3, 4, 6 and 12, while the factors of 18 are 1, 2, 3, 6, 9 and 18. Therefore, HCF(12, 18) = 6. The common positive multiples of 12 and 18 start with 36, so LCM(12, 18) = 36.

HCF and LCM Formula & Tricks

Important Formulas

Product relationship for two numbers
HCF(a, b) × LCM(a, b) = a × b

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

LCM using HCF
LCM(a, b) = (a × b) ÷ HCF(a, b)

Use this formula after finding the HCF of two positive integers.

HCF by prime factorisation
HCF = product of common prime factors with the smallest powers

Choose only the primes present in every number and use their minimum exponents.

LCM by prime factorisation
LCM = product of all prime factors with the greatest powers

Include every prime appearing in the numbers and use its maximum exponent.

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

First express the fractions in lowest terms before applying the formula.

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

Use reduced fractions and simplify the final result if necessary.

Quick Tricks

Use the division algorithm for two large numbers

Divide the larger number by the smaller number, then divide the previous divisor by the remainder. Continue until the remainder becomes zero. The last non-zero remainder is the HCF.

Example: For 252 and 198: 252 = 198 × 1 + 54, 198 = 54 × 3 + 36, 54 = 36 × 1 + 18, and 36 = 18 × 2. Hence, HCF = 18.
Find LCM from the HCF relationship

For two numbers, calculate LCM by dividing their product by the HCF. This is often faster than listing multiples.

Example: For 84 and 126, HCF = 42. Therefore, LCM = (84 × 126) ÷ 42 = 252.
Check coprime numbers quickly

If two numbers have HCF 1, they are coprime and their LCM equals their product.

Example: For 8 and 15, HCF = 1, so LCM = 8 × 15 = 120.
Convert repeated-event wording into LCM

When events occur at fixed intervals and must coincide again, use the LCM of the intervals. If the question asks for the largest equal grouping, use the HCF.

Example: Signals flashing every 12 and 18 seconds flash together again after LCM(12, 18) = 36 seconds.

HCF and LCM Concepts

Prime Factorisation Method

Prime factorisation finds HCF from minimum powers and LCM from maximum powers of the prime factors.

Write each number as a product of primes. For HCF, retain only the primes common to all numbers and take the smallest exponent. For LCM, retain every prime and take the greatest exponent.

Example: 24 = 2³ × 3 and 36 = 2² × 3². Thus, HCF = 2² × 3 = 12 and LCM = 2³ × 3² = 72.

Division Method for HCF

The division method finds the HCF by repeatedly dividing the divisor by the remainder.

For two numbers a and b, where a > b, write a = bq + r. Replace a and b with b and r, and repeat until the remainder is zero. The last non-zero remainder is the HCF. For more than two numbers, find the HCF of two numbers first and then combine it with the next number.

Example: For 455 and 315: 455 = 315 + 140, 315 = 140 × 2 + 35, and 140 = 35 × 4. Therefore, HCF = 35.

Relationship Between HCF and LCM

For two positive integers, HCF × LCM equals the product of the two integers.

If the HCF of two numbers is h and their LCM is l, then h × l = a × b. This relationship applies directly to two numbers, not as a simple product rule for three or more numbers. The HCF always divides the LCM.

Example: If two numbers are 18 and 30, their HCF is 6. Hence, LCM = (18 × 30) ÷ 6 = 90.

Applications in Word Problems

Use LCM for the first common occurrence of repeating intervals and HCF for the largest equal division or grouping.

For repeated events, take the LCM of the time intervals. For cutting lengths into the longest equal pieces, take the HCF of the lengths. For distributing objects into the greatest number of identical groups, the HCF gives the number of groups when each group must contain the same number from every category.

Example: Ropes of 48 m and 60 m are cut into equal pieces of maximum length. The piece length is HCF(48, 60) = 12 m, giving 4 and 5 pieces respectively.

HCF and LCM Video Lessons

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14 Lessons
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HCF and LCM in Number System

Understand the concepts of Highest Common Factor and Least Common Multiple, including how to identify and calculate HCF and LCM in number system problems.

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Practice HCF and LCM Questions

Practise published questions related to this topic.

1√(√400 + √25) = ?→ 2What is the smallest number that is a multiple of 11, 13, 22 and 26?→ 3The largest four-digit number which when divided by 14, 15 and 8 leaves remainder 4 in each case is:→ 4A record entry gives √32 . Simplify.→ 558P8 is divisible by 9. What is the value of P?→ 6What is the largest 5 digit number divisible by 89?→ 7What is the minimum value that must be assigned to A so that the 8-digit number 322A4828 is divisible by 4?→ 8The Highest Common Factor of two numbers is 15 and their Least Common Multiple is 360. What is the product of the two numbers?→ 9The Highest Common Factor (HCF) of two numbers is 48 and their Least Common Multiple (LCM) is 3456. If one of the numbers is 768, find the other number.→ 10The product of two numbers is 5040. If their LCM is 420, find the ratio of their LCM to their HCF.→

HCF and LCM Quick Quiz

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Quick Revision Notes

HCF and LCM Quick Revision

Use these rules to select the correct method and verify calculations.

  • HCF is the greatest common divisor; LCM is the least positive common multiple.
  • In prime factorisation, HCF uses minimum powers of common primes.
  • In prime factorisation, LCM uses maximum powers of all occurring primes.
  • For two positive integers, HCF × LCM = product of the numbers.
  • Use the division algorithm or prime factorisation to find HCF.
  • Use LCM for simultaneous repetition and HCF for greatest equal division.
  • If two numbers are coprime, their HCF is 1 and their LCM is their product.
  • The HCF of two numbers always divides their LCM.

HCF and LCM FAQs

What is the formula for the LCM of two numbers when their HCF is known?

LCM(a, b) = (a × b) ÷ HCF(a, b). For 16 and 24, HCF = 8, so LCM = (16 × 24) ÷ 8 = 48.

How do you find HCF and LCM using prime factors?

Use the smallest powers of common prime factors for HCF and the greatest powers of all prime factors for LCM. For 20 = 2² × 5 and 30 = 2 × 3 × 5, HCF = 2 × 5 = 10 and LCM = 2² × 3 × 5 = 60.

What is the HCF and LCM of two coprime numbers?

Their HCF is 1, and their LCM is the product of the numbers. For 9 and 16, HCF = 1 and LCM = 144.

Which one is used for repeated events, HCF or LCM?

LCM is used to find when repeating events occur together again. For intervals of 8 and 12 minutes, the next common occurrence is after LCM(8, 12) = 24 minutes.

Which one is used to make the largest equal pieces from given lengths?

HCF is used because the piece must be the greatest length that divides every given length exactly. For 36 m and 54 m, the maximum equal piece length is HCF(36, 54) = 18 m.

Can the product relationship be used for three numbers directly?

No. The rule HCF × LCM = product of numbers applies directly to two positive integers. For three or more numbers, calculate HCF and LCM using prime factorisation or successive pairwise operations.

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