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Prime factors, HCF and LCM
Every whole number is built from prime numbers, like a word is built from letters. Once you can see the primes inside a number, HCF and LCM questions become quick.
What is a prime?
A prime number has exactly two factors: 1 and itself.
The first few primes are 2, 3, 5, 7, 11, 13, 17, 19.
1 is not prime, because it only has one factor.
The first few primes are 2, 3, 5, 7, 11, 13, 17, 19.
1 is not prime, because it only has one factor.
Break a number into primes
Split the number into any two factors. Keep splitting until every branch ends in a prime. Circle the primes.
Then write them as a product, using powers for repeats.
60 = 2 × 2 × 3 × 5 = 22 × 3 × 5
Then write them as a product, using powers for repeats.
60 = 2 × 2 × 3 × 5 = 22 × 3 × 5
HCF and LCM
HCF (highest common factor) is the biggest number that divides into both numbers.
LCM (lowest common multiple) is the smallest number that both numbers divide into.
Put the prime factors in a Venn diagram. Shared primes go in the middle.
HCF = multiply the primes in the middle.
LCM = multiply every prime in the diagram.
LCM (lowest common multiple) is the smallest number that both numbers divide into.
Put the prime factors in a Venn diagram. Shared primes go in the middle.
HCF = multiply the primes in the middle.
LCM = multiply every prime in the diagram.
Find the HCF and LCM of 60 and 84.
- 60 = 22 × 3 × 5 and 84 = 22 × 3 × 7
- Shared primes: 2, 2 and 3. These go in the middle.
- HCF = 2 × 2 × 3 = 12
- LCM = 5 × (2 × 2 × 3) × 7 = 420
- Quick check: 60 × 84 = 5040 and 12 × 420 = 5040. They match.
Answer: HCF = 12, LCM = 420
Break both numbers into primes. HCF = product of the shared primes. LCM = product of all the primes in the Venn diagram.
The interactive lesson includes the diagrams for this topic.
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