Common factors are common subparts
A factor of both numbers is a subpart of both factorisations.
Inside both numbers
45 = 3 × 3 × 5 and 75 = 3 × 5 × 5. A factor of 45 is a subpart of 3 × 3 × 5. A factor of 75 is a subpart of 3 × 5 × 5. A common factor must be a subpart of both.
Which subparts do you think are in both factorisations?
What this lesson covers
The idea
The common factors of numbers are the subparts common to all their prime factorisations, together with 1.
Inside both numbers
45 = 3 × 3 × 5 and 75 = 3 × 5 × 5. A factor of 45 is a subpart of 3 × 3 × 5. A factor of 75 is a subpart of 3 × 5 × 5. A common factor must be a subpart of both.
Which subparts do you think are in both factorisations?
- 3, 5 and 3 × 5
- 3 × 3 and 5 × 5
- Only 1
Look for the same beads
Pick a subpart in the top row. If every bead finds its own twin in the bottom row, it is a common factor. Find them all.
Subparts that fit in both
A subpart is common only if both factorisations contain it. Look at 45 = 3 × 3 × 5 and 75 = 3 × 5 × 5:
So the common factors of 45 and 75 are 3, 5 and 15, and 1 (the empty subpart). The highest is 15.
The common factors of numbers are the subparts common to all their prime factorisations, together with 1.
- Subpart | In 45? | In 75?
- 3 | yes | yes
- 5 | yes | yes
- 3 × 5 = 15 | yes | yes
- 3 × 3 = 9 | yes | no: one 3
- 5 × 5 = 25 | no: one 5 | yes
Notes
The common factors of numbers are the subparts common to all their prime factorisations, together with 1.
Check yourself
Which number is a common factor of 12 = 2 × 2 × 3 and 18 = 2 × 3 × 3?
28 = 2 × 2 × 7 and 42 = 2 × 3 × 7. How many common factors do they have? Count 1 as well.
Answer: 4
The common subparts are 2, 7 and 2 × 7 = 14, plus 1. The common factors are 1, 2, 7 and 14: four of them.
96 = 2 × 2 × 2 × 2 × 2 × 3 and 275 = 5 × 5 × 11 share no prime. What are their common factors?
A common factor of 45 and 75 is made of exactly two primes. Which number is it?
Answer: 15
The two-prime subparts of 45 are 3 × 3, 3 × 5 and 5 × 5. Only 3 × 5 = 15 is also a subpart of 75.
- 4. 4 = 2 × 2 needs two 2s, but 18 has only one 2.
- 6 — correct. Yes! 6 = 2 × 3 is a subpart of both factorisations.
- 9. 9 = 3 × 3 needs two 3s, but 12 has only one 3.
- Only 1 — correct. Yes! No prime is in both, so the only common subpart is the empty one: 1.
- None at all. 1 divides every number, so 1 is always a common factor.
- 2 and 5. 2 is not in 275 and 5 is not in 96. A common factor must be in both.