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How to Learn HCF and LCM Quickly and Correctly | Quick Notes

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by

eduKate Asia
Three students sit around open books and worksheets at a classroom table, reading, writing and discussing the work together.

To learn HCF and LCM quickly and correctly, begin by distinguishing two opposite questions. The highest common factor (HCF) is the largest positive whole number that divides all the given numbers exactly. The lowest common multiple (LCM) is the smallest positive whole number that is a multiple of all the given numbers. One is useful for dividing things into the largest equal groups; the other often helps with repeating schedules. These eduKateSG Mathematics Quick Notes explain both ideas with clear examples and checks.

  • HCF: find the largest number that divides every given number exactly.
  • LCM: find the smallest positive number each given number divides exactly.
  • Start visually: list factors for HCF or multiples for LCM.
  • Use prime factors: choose the smaller shared exponent for HCF and the largest required exponent for LCM.
  • Match the story: equal biggest groups suggest HCF; first shared repetition suggests LCM.
  • Check: HCF divides every number; every number divides the LCM.

First, find an HCF by listing the positive factors. For 12, they are 1, 2, 3, 4, 6 and 12. For 18, they are 1, 2, 3, 6, 9 and 18. The common factors are 1, 2, 3 and 6. The largest is 6, so HCF(12, 18) = 6. You can check: 12 ÷ 6 = 2 and 18 ÷ 6 = 3. Both give whole-number answers. The HCF cannot be larger than the smaller of the two positive numbers.

Second, find an LCM by listing positive multiples. Multiples of 12 include 12, 24, 36, 48, …, while multiples of 18 include 18, 36, 54, …. The first shared positive multiple is 36, so LCM(12, 18) = 36. Check: 36 ÷ 12 = 3 and 36 ÷ 18 = 2. Listing works well with small numbers, but becomes slow when the numbers are large. In that case, prime factorisation gives a cleaner route.

Third, use prime factors to solve both at once. Let 24 = 2³ × 3 and 36 = 2² × 3². For the HCF, take each prime that appears in both numbers to the smaller exponent: 2² × 3 = 12. For the LCM, take every prime needed to build either number, using the larger exponent: 2³ × 3² = 72. The smaller exponent gives only factors that both numbers contain; the larger exponent provides enough factors for both numbers to divide the multiple exactly.

Worked example: 18 and 30

Write 18 = 2 × 3² and 30 = 2 × 3 × 5. The common primes with their smaller exponents are 2 and 3, so HCF = 2 × 3 = 6. For the LCM, use 2, 3² and 5, giving LCM = 2 × 9 × 5 = 90. Check: 18 ÷ 6 = 3 and 30 ÷ 6 = 5; also 90 ÷ 18 = 5 and 90 ÷ 30 = 3. Both checks work.

Story problem: largest equal packets

A teacher has 24 red stickers and 36 blue stickers and wants to make as many identical packs as possible, using every sticker. Each pack must contain the same number of red stickers and the same number of blue stickers. The number of packs must divide both 24 and 36 exactly, so use the HCF = 12. The teacher makes 12 packs, each containing 24 ÷ 12 = 2 red stickers and 36 ÷ 12 = 3 blue stickers. The story’s goal is the greatest possible number of equal packs.

Story problem: two repeating events

Two indicator lights flash together at the beginning. One flashes every 8 seconds and the other every 12 seconds. Assuming their schedules continue unchanged, when will they next flash together? List the times: first light at 8, 16, 24; second at 12, 24. The first common positive time is 24 seconds, so LCM(8, 12) = 24. This is an LCM question because you are looking for the earliest shared repetition, not trying to divide a fixed number of objects.

HCF or LCM? Ask the meaning first

  • Largest possible equal group size: usually HCF, if the group size must divide each quantity exactly.
  • Greatest number of identical packs using all items: usually HCF, when the pack count divides every original quantity.
  • Smallest positive total divisible by several pack sizes: LCM.
  • Next time cycles happen together: LCM when cycles start together and repeat at fixed intervals.
  • Question asks “greatest”: not automatically HCF; identify what must divide what.
  • Question asks “least”: not automatically LCM; check the actual relationship.

The useful check for two positive integers

For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b. With 18 and 30, the left side is 6 × 90 = 540; the right side is 18 × 30 = 540. This is a handy check for two positive numbers. Do not blindly use the same product identity for three or more numbers, because it does not generally hold in that simple form.

What happens with more than two numbers?

Suppose the numbers are 12, 18 and 30. Their prime factorisations are 2² × 3, 2 × 3² and 2 × 3 × 5. For HCF, take the minimum exponent shared across all three: 2 × 3 = 6. For LCM, take the maximum exponent required: 2² × 3² × 5 = 180. Check that 6 divides 12, 18 and 30, and that all three numbers divide 180 exactly. The prime-factor method extends naturally to three or more positive integers.

Common mistakes that spoil the answer

  • Choosing the biggest factor in just one number: HCF must divide every number.
  • Choosing the first multiple of one number: LCM must be divisible by every number.
  • Using largest exponents for HCF: those exponents usually belong to LCM.
  • Leaving out a prime that occurs in only one number for LCM: include every required prime.
  • Reading only the words “greatest” or “lowest”: translate the situation into divisibility.
  • Not checking: test all required divisions in the original numbers.

Mini-test: HCF, LCM and reasoning

  1. Find the HCF of 20 and 28.
  2. Find the LCM of 6 and 15.
  3. Find the HCF and LCM of 16 and 24.
  4. Two bells ring together now. One repeats every 9 minutes and the other every 12 minutes. When will they next ring together?

Answers with quick checks

1. HCF(20, 28) = 4. 2. LCM(6, 15) = 30. 3. 16 = 2⁴ and 24 = 2³ × 3, giving HCF = 8 and LCM = 48. Check 8 × 48 = 16 × 24 = 384. 4. LCM(9, 12) = 36 minutes, so they next ring together in 36 minutes.

A ten-minute HCF and LCM study routine

  1. Minutes 1–2: list factors of two small numbers and circle the common factors.
  2. Minutes 3–4: list their positive multiples and identify the first common one.
  3. Minutes 5–7: prime-factorise a new pair and calculate both HCF and LCM.
  4. Minute 8: verify the results with division.
  5. Minutes 9–10: invent one equal-packets question and one repeating-events question, then say which method each needs.

When you can explain why one problem needs a common divisor and another needs a common multiple, HCF and LCM stop looking like two arbitrary formulas. The words of the question tell you which mathematical structure to use.

Learn the next connected idea

Use these eduKateSG Mathematics Quick Notes to build on today’s learning: Prime Factorisation · Factors and Multiples · undefined · Mathematics Word Problems. Complete a fresh question from memory before moving on.

The longer teaching reference, Secondary 1 Mathematics Tutor Clementi | Small Groups Tutorials, remains unchanged. This is a separate Quick Notes lesson.