HCF, LCM, prime numbers and divisibility game
A maths game for really understanding the highest common factor and the lowest common multiple, instead of memorising a recipe. Each number appears as a row of coloured blocks, one per prime factor: 2 is blue, 3 green, 5 orange and 7 purple, so 12 is two blue blocks and one green. There are no answers to pick from: the HCF and LCM are built by adding and removing blocks of each colour, numbers are broken by dividing them by primes (with a 4 and a 9 as traps), and the multiples and primes are tapped among eight numbers. The rows of two numbers are lined up by colour, and when you answer the blocks that count light up: for the HCF, the ones both numbers have; for the LCM, for each colour, the longest row. There are also divisibility rules, numbers that look prime but are not, factorisations with a trap and real-life word problems where you have to decide whether you need an LCM or an HCF.
Mathematics · Primary (Years 5-6), Secondary
“Common factors with the lowest power” and “all factors with the highest power” are phrases that get forgotten and mixed up. With the blocks lined up you see why: the HCF has to fit into both numbers, so it can only use the blocks both of them have; the LCM has to contain both, so it needs all of them.
Wrong answers are not random: giving the LCM when the HCF is asked, multiplying the two numbers, writing a 4 or a 9 inside a factorisation (the value matches, but they are not prime), thinking every multiple of 3 is a multiple of 9, or that ending in 5 is enough to divide by 10. Each mistake comes with its explanation.
Buses that leave together again, lighthouses that flash at the same time, ropes cut into equal pieces or a floor covered with square tiles. Working out what is being asked is the hardest part in exams, and the explanation always says it: if things coincide again it is a multiple; if you cut into equal pieces as big as possible, a factor.
"One bus leaves every 12 minutes and another every 18. They have just left together: when will they coincide again? We need a multiple of both, the smallest one: an LCM. 12 = 2² · 3 and 18 = 2 · 3². In blocks, 12 has two blue and one green, and 18 one blue and two green. The LCM takes, for each colour, the longest row: two blue and two green, 2² · 3² = 36. They leave together again in 36 minutes. Multiplying 12 · 18 = 216 would also be a time they coincide, but not the first one."
Prefer to learn without screens? Try this analogue version:
Cut out card squares in four colours and write 2 on the blue ones, 3 on the green, 5 on the orange and 7 on the purple.
Factorise a number (for example, 60) by dividing by primes and lay down one block per factor: 2, 2, 3, 5.
Do the same with another number underneath, lining up the colours, and set aside the blocks both have: that is the HCF.
Put all the blocks together without repeating the shared ones and multiply: that is the LCM. Check that HCF × LCM = the product of the two numbers.
If you like Prime Blocks, these activities work similar skills:
Another maths game: equivalent fractions and simplifying, which is exactly where the HCF is used in practice.
Write the numbers from 1 to 100 in a grid and cross out the multiples of 2, 3, 5 and 7: the ones left (apart from 1) are the primes.
Knowing the multiplication tables well is what makes factorising quick: if you know 7 × 8 = 56, you know 56 is divisible by 7.
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