Lowest common multiple and highest common factor — Mathematics, 11–13
Find the LCM and HCF of two numbers with lists or prime factors, choose the quickest method and use them to solve problems.
The smallest shared multiple and the largest shared divisor
The lowest common multiple (LCM) of two numbers is the smallest multiple they share; the highest common factor (HCF) is the largest divisor they share. Take 8 and 12. Multiples of 8: {8, 16, 24, 32, …}; of 12: {12, 24, 36, …}. The first shared one is 24, so LCM(8, 12) = 24. Divisors of 8: {1, 2, 4, 8}; of 12: {1, 2, 3, 4, 6, 12}. Shared: {1, 2, 4}, the largest is 4, so HCF(8, 12) = 4. For the LCM you can stop listing at the product 8 × 12 = 96, which is already a common multiple.
Why prime factors give the answer without lists
Factorise: 12 = 2² × 3 and 18 = 2 × 3². A common multiple must contain the factors of both numbers, so the LCM takes every prime with its greater exponent: 2² × 3² = 36. A common divisor can only contain what both numbers have, so the HCF takes only the shared primes with their smaller exponent: 2 × 3 = 6. Check: 36 ÷ 12 = 3 and 36 ÷ 18 = 2; 12 ÷ 6 = 2 and 18 ÷ 6 = 3.
84 and 90: choosing the method, then calculating
With lists, the LCM of 84 and 90 could send you as far as 84 × 90 = 7560: not practical (with 8 and 12, lists are quicker). So factorise: 84 = 2² × 3 × 7 and 90 = 2 × 3² × 5. LCM: all the primes, greater exponents: 2² × 3² × 5 × 7 = 4 × 9 × 5 × 7 = 1260. HCF: shared primes (2 and 3), smaller exponents: 2 × 3 = 6. Check: 1260 ÷ 84 = 15 and 1260 ÷ 90 = 14; 84 ÷ 6 = 14 and 90 ÷ 6 = 15.
The trap: always multiplying the two numbers
A reasonable slip: taking the product as the LCM. The product is a common multiple, but not always the smallest. So look at the numbers before calculating. If one is a multiple of the other, the LCM is the larger and the HCF is the smaller: LCM(30, 6) = 30 and HCF(30, 6) = 6, not 180. If one is prime, either it divides the other and is the HCF (HCF(7, 21) = 7, LCM = 21), or it does not, and then HCF(15, 7) = 1 and LCM = 15 × 7 = 105.
Where you see it outside school: meeting again and cutting evenly
Two signal lights flash, one every 12 seconds and the other every 18. They flash together every 36 seconds: LCM(12, 18). A rectangular sheet 360 mm by 150 mm is to be cut into equal squares, as big as possible, with no waste: the side is HCF(360, 150) = 30 mm, and you get 12 × 5 = 60 squares. The LCM shows up when something repeats and you want the next meeting; the HCF when you share or cut into equal parts as large as possible.
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