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Unit I · Real numbers
8 min readPathway sheet1-mark MCQ, 2-mark and 3-mark questions

HCF and LCM by prime factorisation

Write each number as a product of primes. HCF = product of the smallest powers of common primes; LCM = product of the greatest powers of all primes.

HCFProduct of the smallest power of each common prime factor
LCMProduct of the greatest power of each prime factor that appears
Two numbers a and bHCF(a, b) × LCM(a, b) = a × b
Three numbersThe product rule does not hold in general

What HCF and LCM mean

The HCF (highest common factor) is the biggest number that divides each given number exactly. The LCM (lowest common multiple) is the smallest number that each given number divides exactly. The HCF is never bigger than the smallest number. The LCM is never smaller than the biggest number. The HCF always divides the LCM.

Method: HCF and LCM by prime factorisation

  1. 1
    Write each number as a product of primes, using powers.
  2. 2
    List every prime that appears in any of the numbers.
  3. 3★
    HCF: take only the primes common to all the numbers, each with its smallest power. Multiply.
  4. 4★
    LCM: take every prime, each with its greatest power. Multiply.
  5. 5
    For two numbers, check that HCF × LCM equals the product of the numbers.

★ key step

Example 1: HCF and LCM of 84 and 120

Problem

Find the HCF and LCM of 84 and 120 by prime factorisation. Verify that HCF × LCM = product of the numbers.

Working
  1. 84 = 2² × 3 × 7
  2. 120 = 2³ × 3 × 5
  3. Common primes: 2 and 3. Smallest powers: 2² and 3¹.
  4. HCF = 2² × 3 = 12
  5. All primes with greatest powers: 2³, 3¹, 5¹, 7¹.
  6. LCM = 2³ × 3 × 5 × 7 = 840
  7. HCF × LCM = 12 × 840 = 10080
  8. 84 × 120 = 10080. The two products are equal.
Answer

HCF = 12, LCM = 840, and 12 × 840 = 84 × 120 = 10080.

Example 2: three numbers

Problem

Find the HCF and LCM of 12, 18 and 30. Does HCF × LCM equal 12 × 18 × 30?

Working
  1. 12 = 2² × 3, 18 = 2 × 3², 30 = 2 × 3 × 5
  2. Common primes in all three: 2 and 3, each with smallest power 1.
  3. HCF = 2 × 3 = 6
  4. Greatest powers: 2², 3², 5¹.
  5. LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180
  6. HCF × LCM = 6 × 180 = 1080
  7. 12 × 18 × 30 = 6480, which is not 1080.
Answer

HCF = 6, LCM = 180. The product rule does not work for these three numbers.

Example 3: using the product rule

Problem

The HCF of two numbers is 18 and their product is 12960. Find their LCM.

Working
  1. Rule for two numbers: HCF × LCM = product of the numbers.
  2. 18 × LCM = 12960
  3. LCM = 12960 ÷ 18 = 720
  4. Check with a real pair: 144 × 90 = 12960, HCF(144, 90) = 18 and LCM(144, 90) = 720.
Answer

LCM = 720

In real life

A shopkeeper uses the HCF to pack two kinds of sweets in the largest equal boxes with nothing left over. A bus depot uses the LCM to find when two routes leave together again.

Exam angle

1 marks: “Given HCF(26, 169) = 13, find LCM(26, 169).” LCM = (26 × 169) ÷ 13 = 4394 ÷ 13 = 338.

2 marks: “Find the HCF and LCM of 90 and 144 by prime factorisation.” 90 = 2 × 3² × 5 and 144 = 2⁴ × 3². HCF = 2 × 3² = 18. LCM = 2⁴ × 3² × 5 = 720.

3 marks: “Find the HCF and LCM of 24, 36 and 60 by prime factorisation.” 24 = 2³ × 3, 36 = 2² × 3², 60 = 2² × 3 × 5. HCF = 2² × 3 = 12. LCM = 2³ × 3² × 5 = 360.

Remember it

HCF is small and shared: smallest powers of the common primes. LCM is large and complete: greatest powers of every prime.

Read alongside the books on your syllabus: NCERT, Mathematics: Textbook for Class X · NCERT, Exemplar Problems: Mathematics, Class X. Written from standard textbooks; faculty review in progress. Spotted an error? Report it.