HCF from the primes
Each common prime, as few times as it appears.
Which primes go in?
30 = 2 × 3 × 5 and 72 = 2 × 2 × 2 × 3 × 3. The HCF is the largest common subpart of the two factorisations.
Look at the prime 5. Should 5 be in the HCF of 30 and 72?
What this lesson covers
The idea
The HCF is the largest common subpart: take each prime common to all the factorisations the minimum number of times it appears in them; this works for two or more numbers.
Which primes go in?
30 = 2 × 3 × 5 and 72 = 2 × 2 × 2 × 3 × 3. The HCF is the largest common subpart of the two factorisations.
Look at the prime 5. Should 5 be in the HCF of 30 and 72?
- Yes, 30 has a 5
- No, 72 has no 5
- Yes, if it is at the end
Build the HCF
For each prime, set how many beads go in the answer. The toy checks every number and tells you what is wrong.
The smaller count, every time
A prime can be in the HCF only if it is in every number. And it can appear only as many times as it appears in the number that has fewer.
So the HCF of 30 and 72 is 2 × 3 = 6. For 225 = 3 × 3 × 5 × 5 and 750 = 2 × 3 × 5 × 5 × 5: one 3, two 5s, so the HCF is 3 × 5 × 5 = 75.
The HCF is the largest common subpart of the prime factorisations: take each prime common to all the numbers the minimum number of times it appears in them. This works for two or more numbers.
- Prime | Times in 30 and 72 | HCF takes
- 2 | 1 and 3 | 1
- 3 | 1 and 2 | 1
- 5 | 1 and 0 | 0
Notes
The HCF is the largest common subpart: take each prime common to all the numbers the minimum number of times it appears. It works for two or more numbers.
Check yourself
Find the HCF of 84 = 2 × 2 × 3 × 7 and 126 = 2 × 3 × 3 × 7.
Answer: 42
2: one in 126, two in 84, so take one. 3: one in 84, two in 126, so take one. 7: one in each. HCF = 2 × 3 × 7 = 42.
Which is the HCF of 2 × 2 × 2 × 3 × 5 (= 120) and 2 × 2 × 3 × 3 × 7 (= 252)?
Find the HCF of three numbers: 12, 18 and 30.
Answer: 6
The primes in all three are 2 and 3. Each appears at least once in each number, so the HCF is 2 × 3 = 6.
35 = 5 × 7 and 66 = 2 × 3 × 11 share no prime. What is their HCF?
- 2 × 2 × 2 × 3 × 3 = 72. That takes the larger count of 2s and 3s. The HCF takes the smaller counts.
- 2 × 2 × 3 = 12 — correct. Yes! Two 2s (the smaller count) and one 3 (the smaller count). 5 and 7 are in only one number each.
- 2 × 3 × 5 × 7 = 210. That uses every prime, but 5 and 7 are in only one of the numbers.
- There is no HCF. 1 divides both numbers, so it is a common factor. The highest one is 1.
- 5. 5 is not a factor of 66.
- 1 — correct. Yes! The only common subpart is the empty one, which is 1.