Share only what both have

The idea

In the prime factorisation method, the HCF of given numbers is the product of the smallest power of each prime factor common to the numbers.

6 and 20

Write both numbers as prime beads: 6 = 2 × 3 and 20 = 2² × 5.

Which primes do both numbers have?

• Only 2
• 2, 3 and 5
• None

Take what both can give

For each prime, take the beads that every number has. Use + and −, then check your HCF.

Smallest power of each common prime

The HCF is the product of the smallest power of each prime factor that is common to the numbers.

So HCF(6, 20) = 2^1 = 2. For 96 = 2^5 × 3 and 404 = 2^2 × 101, only 2 is common and the smaller power is 2^2, so HCF = 4.

• Prime | In 6 | In 20 | Common? Smallest power
• 2 | 2^1 | 2^2 | Yes: 2^1
• 3 | 3^1 | none | No
• 5 | none | 5^1 | No

Notes

The HCF is the product of the smallest power of each prime factor common to the numbers. Example: HCF(6, 20) = 2^1 = 2.

Check yourself

Find the HCF of 60 = 2² × 3 × 5 and 84 = 2² × 3 × 7.

Answer: 12

Common primes are 2 and 3. Smallest powers: 2^2 and 3^1. HCF = 4 × 3 = 12.

12 = 2² × 3 and 18 = 2 × 3². What is the HCF?

15 = 3 × 5 and 28 = 2² × 7. What is the HCF?

Find the HCF of 6, 72 and 120, given 6 = 2 × 3, 72 = 2³ × 3², 120 = 2³ × 3 × 5.

Answer: 6

2 and 3 are in all three. Smallest powers: 2^1 and 3^1. HCF = 2 × 3 = 6.

• 2² × 3² = 36. That uses the greatest power of each prime. The HCF uses the smallest.
• 2 × 3 = 6 — correct. Yes! Smallest power of 2 is 2^1 and smallest power of 3 is 3^1. HCF = 6.
• 2² × 3 = 12. The number 18 has only one 2. Take the smaller power, 2^1.
• 1. They do share primes: both have 2 and 3.
• 1 — correct. Yes! They have no prime in common, so nothing to take: the HCF is 1.
• 420. That is 15 × 28, not the HCF. The HCF takes only primes that are in both.
• 3. 3 is only in 15. 28 has no 3.
• 7. 7 is only in 28. 15 has no 7.