Step by step: way 1, prime factors
The question
Question: find the greatest common factor of 45 and 60.
GCF(45, 60) = ?
A common factor goes into both numbers.
The greatest common factor, the GCF, is the biggest one.
common factor: goes into both
GCF: the greatest one
1 Find the prime factors
3 goes into 45.
45 is 3 times 15.
45 = 3 × 15
3 goes into 15.
15 is 3 times 5.
5 is prime, so we stop.
= 3 × 3 × 5
60 is even, so 2 goes in.
60 is 2 times 30.
60 = 2 × 30
30 is even, so 2 goes in.
30 is 2 times 15.
= 2 × 2 × 15
3 goes into 15.
15 is 3 times 5.
5 is prime, so we stop.
= 2 × 2 × 3 × 5
2 Multiply the shared primes
Both have a 3.
Both have a 5.
Each shared prime goes into both numbers.
Together they make the biggest common factor.
We multiply them.
GCF(45, 60) = 3 × 5
3 times 5 is 15.
3 × 5 = 15
So the GCF is 15.
= 15
Careful: the GCF goes into every number.
So it is never bigger than the smallest one.
Careful: GCF ≤ the smallest number
3 Check
Let's check: 15 goes into each number.
45 ÷ 15 = 3
60 ÷ 15 = 4
Nothing is left over.
So 15 is right!
Let's look back.
First: find the prime factors.
Then: multiply the shared primes.
Last: check.
Step by step: way 2, lists
The question
Question: find the greatest common factor of 45 and 60.
GCF(45, 60) = ?
A common factor goes into both numbers.
The greatest common factor, the GCF, is the biggest one.
common factor: goes into both
GCF: the greatest one
1 List the factors
The factors of 45 are 1, 3, 5, 9, 15 and 45.
Factors of 45:1, 3, 5, 9, 15, 45
They come in pairs: 1 times 45, 3 times 15, 5 times 9.
1 × 45, 3 × 15, 5 × 9
The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.
Factors of 60:1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
They come in pairs: 1 times 60, 2 times 30, 3 times 20, 4 times 15, 5 times 12, 6 times 10.
1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12, 6 × 10
2 Find the common ones
1, 3, 5 and 15 are in both lists.
These are the common factors.
Common:1, 3, 5, 15
3 Pick the greatest
The greatest one is 15.
So the GCF is 15.
GCF(45, 60) = 15
Careful: the GCF goes into every number.
So it is never bigger than the smallest one.
Careful: GCF ≤ the smallest number
4 Check
Let's check: 15 goes into each number.
45 ÷ 15 = 3
60 ÷ 15 = 4
Nothing is left over.
So 15 is right!
Let's look back.
First: list the factors.
Then: find the common ones.
Then: pick the greatest.
Last: check.
