Prime factors, HCF × LCM, and exact roots
Split two numbers into primes with factor trees. Build the HCF and LCM from the primes, and find square and cube roots by pairing and grouping the primes in threes.
- UK: Year 8
- France: grade 8
- Germany: grade 5
- Iran: grade 7
- Afghanistan: grade 7
- Syria: grade 5
- Palestine: grade 5
- Ukraine: grade 6
- Russia: grade 7
Guide
This studio follows Year 8 (KS3 "prime factorisation", "integer powers and associated real roots"). Every composite number splits into primes with a factor tree; this is its prime factorisation, and the order you split in does not change the primes you end with.
highest common factor (HCF): multiply the primes that both numbers have, as many times as both have them. lowest common multiple (LCM): take each prime the greatest number of times it appears in either number. For two numbers, HCF × LCM = the product of the numbers.
square root: put the primes in pairs and take one from each pair. cube root: put them in groups of three and take one from each group.
Common mistakes: thinking the LCM is always a × b; using HCF × LCM = product for three numbers; splitting the root of a sum (√(9 + 16) ≠ 3 + 4); thinking 2 is not prime because it is even (it is the only even prime); and mixing up "9 has two square roots, 3 and −3" with √9, which is 3 only.