The HCF, the LCM and the product
Take 36 and 60. Their HCF is 12 and their LCM is 180. Multiply those two: 12 × 180 = 2160.
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- 36 and 60, each written as primes.
- The pairs 2, 2 and 3 make the HCF, 12.
- A 3 and a 5 are left over.
- The LCM is the pairs and the left-overs: 12 × 3 × 5 = 180.
- 12 × 180 = 2160, and 36 × 60 = 2160.
The rule
HCF × LCM equals the product of the two numbers.
Pieces with no partner are the left-overs. Both numbers hold the pairs. So the product counts the pairs twice, and each left-over once. The HCF holds the pairs once. The LCM holds the pairs once, plus the left-overs. Together, HCF × LCM holds the same pieces.
Two co-prime numbers have HCF 1. So their LCM is just their product.
Find the other number
HCF 3, LCM 42, and one number is 6.
- HCF × LCM = 3 × 42 = 126.
- The other number is 126 ÷ 6 = 21.
- Check: HCF(6, 21) = 3 and LCM(6, 21) = 42.
The check can fail. Try HCF 3, LCM 42 and one number 9. The rule gives 126 ÷ 9 = 14. But HCF(9, 14) is 1, not 3. So no such pair exists.
Watch out!
The rule is for two numbers. With three numbers it can fail. For 2, 4 and 8, HCF × LCM is 16. The product is 64.