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Mathématiques

HCF and LCM Practice Questions, By Prime Factors and By Venn

Twelve questions worked two ways, by prime factorisation and by Venn diagram, ending with bus and tiling problems where you must decide which one is wanted.

La réponse en bref

The HCF of two numbers is the product of the prime factors they share; the LCM is the product of every prime factor present, each at its highest power. With 60 = 2² × 3 × 5 and 72 = 2³ × 3², the HCF is 12 and the LCM is 360.

Les exercices

  1. Exercice 1Base2 points

    Find the HCF of 12 and 18.

    Afficher la correction

    6

    1. 12 = 2² × 3 and 18 = 2 × 3².
    2. Shared primes: one 2 and one 3.
    3. In a Venn diagram the overlap contains 2 and 3; 12 keeps a spare 2 and 18 keeps a spare 3.
    4. HCF = 2 × 3 = 6.
  2. Exercice 2Base2 points

    Find the LCM of 12 and 18.

    Afficher la correction

    36

    1. 12 = 2² × 3 and 18 = 2 × 3².
    2. Take each prime to its highest power: 2² and 3².
    3. LCM = 4 × 9 = 36.
    4. Check: 6 × 36 = 216 and 12 × 18 = 216.
  3. Exercice 3Base2 points

    Find the HCF of 24 and 36.

    Afficher la correction

    12

    1. 24 = 2³ × 3 and 36 = 2² × 3².
    2. Shared: 2² and 3, taking the lower power of each.
    3. HCF = 4 × 3 = 12.
    4. Sanity check: 12 divides both 24 and 36, and nothing larger does.
  4. Exercice 4Base2 points

    Find the LCM of 8 and 12.

    Afficher la correction

    24

    1. 8 = 2³ and 12 = 2² × 3.
    2. Highest power of 2 is 2³; highest power of 3 is 3.
    3. LCM = 8 × 3 = 24.
    4. Multiplying 8 × 12 = 96 would also be a common multiple, but it is not the lowest.
  5. Exercice 5Standard4 points

    Find the HCF and the LCM of 60 and 72.

    Afficher la correction

    HCF 12, LCM 360

    1. 60 = 2² × 3 × 5 and 72 = 2³ × 3².
    2. Venn overlap: 2, 2, 3. Left only: 5. Right only: 2, 3.
    3. HCF = 2² × 3 = 12.
    4. LCM = 2³ × 3² × 5 = 8 × 9 × 5 = 360.
    5. Check: 12 × 360 = 4,320 = 60 × 72.
  6. Exercice 6Standard4 points

    Find the HCF and the LCM of 84 and 120.

    Afficher la correction

    HCF 12, LCM 840

    1. 84 = 2² × 3 × 7 and 120 = 2³ × 3 × 5.
    2. Overlap: 2, 2, 3. Left only: 7. Right only: 2, 5.
    3. HCF = 2² × 3 = 12.
    4. LCM = 2³ × 3 × 5 × 7 = 840.
    5. Check: 12 × 840 = 10,080 = 84 × 120.
  7. Exercice 7Standard4 points

    Find the LCM of 15, 20 and 25.

    Afficher la correction

    300

    1. 15 = 3 × 5, 20 = 2² × 5, 25 = 5².
    2. Collect every prime that appears anywhere: 2, 3 and 5.
    3. Take the highest power of each: 2², 3, 5².
    4. LCM = 4 × 3 × 25 = 300.
    5. The product check does not apply to three numbers, so verify by dividing: 300 ÷ 15 = 20, ÷ 20 = 15, ÷ 25 = 12.
  8. Exercice 8Standard4 points

    Find the HCF and the LCM of 96 and 144.

    Afficher la correction

    HCF 48, LCM 288

    1. 96 = 2⁵ × 3 and 144 = 2⁴ × 3².
    2. HCF takes the lower power each time: 2⁴ × 3 = 16 × 3 = 48.
    3. LCM takes the higher power each time: 2⁵ × 3² = 32 × 9 = 288.
    4. Check: 48 × 288 = 13,824 = 96 × 144.
  9. Exercice 9Approfondi4 points

    One bus leaves the depot every 18 minutes and another every 24 minutes. Both leave at 09:00. When do they next leave together?

    Afficher la correction

    10:12

    1. The question asks when both timetables reach the same moment, so this is an LCM.
    2. 18 = 2 × 3² and 24 = 2³ × 3.
    3. LCM = 2³ × 3² = 72 minutes.
    4. 72 minutes after 09:00 is 10:12.
    5. Sanity check: 72 is larger than both 18 and 24, as an LCM must be.
  10. Exercice 10Approfondi5 points

    A floor measuring 315 cm by 450 cm is to be covered exactly by identical square tiles with whole-centimetre sides, using as few tiles as possible. Find the tile size and the number of tiles.

    Afficher la correction

    45 cm tiles, 70 of them

    1. The tile must fit both dimensions exactly, and the fewest tiles means the largest tile — so this is an HCF.
    2. 315 = 3² × 5 × 7 and 450 = 2 × 3² × 5².
    3. Shared: 3² and 5, so HCF = 9 × 5 = 45 cm.
    4. 315 ÷ 45 = 7 tiles along one side; 450 ÷ 45 = 10 along the other.
    5. 7 × 10 = 70 tiles.
  11. Exercice 11Approfondi4 points

    Two ribbons measure 84 cm and 126 cm. They are cut into equal pieces as long as possible, with nothing left over. How long is each piece, and how many pieces are there altogether?

    Afficher la correction

    42 cm, 5 pieces

    1. The piece length must divide both ribbons, and it is to be as long as possible — HCF again.
    2. 84 = 2² × 3 × 7 and 126 = 2 × 3² × 7.
    3. Shared: 2, 3 and 7, so HCF = 42 cm.
    4. 84 ÷ 42 = 2 pieces and 126 ÷ 42 = 3 pieces.
    5. 2 + 3 = 5 pieces in total.
  12. Exercice 12Approfondi5 points

    Two numbers are a and 60. Their HCF is 12 and their LCM is 180. Find a.

    Afficher la correction

    a = 36

    1. For any pair, HCF × LCM = the product of the two numbers.
    2. 12 × 180 = 2,160, so a × 60 = 2,160.
    3. a = 2,160 ÷ 60 = 36.
    4. Check: 36 = 2² × 3² and 60 = 2² × 3 × 5, so the HCF is 2² × 3 = 12.
    5. And the LCM is 2² × 3² × 5 = 180, as required.

Où l'on se trompe

  • Multiplying the two numbers together to get the LCM, which is only right when their HCF is 1.
  • Reversing the rule and taking the highest power for the HCF and the lowest for the LCM.
  • Adding the numbers in the Venn overlap instead of multiplying them.
  • Choosing the LCM for a largest-tile or longest-piece question because the numbers in it are large.
  • Building the LCM from the overlap and one circle only, so a prime that appears in just one of the numbers is left out.

The Venn diagram is the prime factorisation, drawn

Put the prime factors of the first number in the left circle and those of the second in the right, with the shared ones in the overlap. Multiply what is in the overlap and you have the HCF. Multiply everything in all three regions and you have the LCM. Nothing is added at any point.

For 84 = 2 × 2 × 3 × 7 and 120 = 2 × 2 × 2 × 3 × 5, the overlap holds 2, 2 and 3; the left-only region holds 7; the right-only region holds 2 and 5. HCF = 2 × 2 × 3 = 12. LCM = 12 × 7 × 2 × 5 = 840.

The two methods below are not alternatives to choose between. The prime factorisation is faster once it is fluent; the Venn shows a pupil why the answers are what they are, and it makes a forgotten prime visible instead of silent.

Deciding whether a worded question wants HCF or LCM

Ignore the size of the numbers and ask what the answer is a measurement of. If it is a size that has to fit into both quantities — the tile, the length of ribbon, the number of identical bags — it is the HCF, and the answer will be smaller than either number. If it is a moment or a quantity that both quantities reach — the next time two buses coincide, the smallest number of tiles, when two lights flash together — it is the LCM, and the answer will be larger than either number.

A quick sanity check catches most errors: an HCF that is bigger than either number, or an LCM smaller than either, is wrong before you check anything else.

The check that costs one multiplication

For any two whole numbers, HCF × LCM equals the product of the numbers themselves. With 60 and 72: 12 × 360 = 4,320, and 60 × 72 = 4,320. If those two do not match, one of the answers is wrong and you know it without redoing the factorisation.

This works only for a pair. Three numbers have no such shortcut, which is why the LCM of 15, 20 and 25 has to be built directly from the primes.

Questions fréquentes

How do I know whether a word problem wants the HCF or the LCM?

Ask whether the answer has to fit inside both quantities or be reached by both. A tile that fits both floor dimensions, or a piece length that divides both ribbons, is an HCF and comes out smaller than either number. Two buses meeting again, or two lights flashing together, is an LCM and comes out larger.

Does the HCF × LCM check work for three numbers?

No. For a pair, HCF × LCM equals the product of the two numbers, and that is a genuinely useful check. For three or more numbers the relationship breaks down, so an LCM of 15, 20 and 25 has to be built directly from the primes and verified by dividing the answer by each number.

Is listing multiples an acceptable method?

For small numbers, yes, and it is how the idea should be introduced. It stops being practical quickly: finding the LCM of 84 and 120 by listing means writing out multiples past 800. Prime factorisation gets the same answer in three lines and keeps working when the numbers get large.

Why draw a Venn diagram when prime factorisation is quicker?

Because it makes an omission visible. The most common LCM error is forgetting a prime that appears in only one number, and in a Venn that prime is sitting alone in a circle, unmissable. Once a pupil stops making that error the diagram can go; until then it is doing real work.

Dernière mise à jour

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