Step by step: way 1, lists
The question
Question: find the greatest common factor of 24 and 36.
GCF(24, 36) = ?
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 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
Factors of 24:1, 2, 3, 4, 6, 8, 12, 24
They come in pairs: 1 times 24, 2 times 12, 3 times 8, 4 times 6.
1 × 24, 2 × 12, 3 × 8, 4 × 6
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.
Factors of 36:1, 2, 3, 4, 6, 9, 12, 18, 36
They come in pairs: 1 times 36, 2 times 18, 3 times 12, 4 times 9, 6 times 6.
1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6
2 Find the common ones
1, 2, 3, 4, 6 and 12 are in both lists.
These are the common factors.
Common:1, 2, 3, 4, 6, 12
3 Pick the greatest
The greatest one is 12.
So the GCF is 12.
GCF(24, 36) = 12
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: 12 goes into each number.
24 ÷ 12 = 2
36 ÷ 12 = 3
Nothing is left over.
So 12 is right!
Let's look back.
First: list the factors.
Then: find the common ones.
Then: pick the greatest.
Last: check.
Step by step: way 2, prime factors
The question
Question: find the greatest common factor of 24 and 36.
GCF(24, 36) = ?
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
24 is even, so 2 goes in.
24 is 2 times 12.
24 = 2 × 12
12 is even, so 2 goes in.
12 is 2 times 6.
= 2 × 2 × 6
6 is even, so 2 goes in.
6 is 2 times 3.
3 is prime, so we stop.
= 2 × 2 × 2 × 3
36 is even, so 2 goes in.
36 is 2 times 18.
36 = 2 × 18
18 is even, so 2 goes in.
18 is 2 times 9.
= 2 × 2 × 9
3 goes into 9.
9 is 3 times 3.
3 is prime, so we stop.
= 2 × 2 × 3 × 3
2 Multiply the shared primes
Both have a 2.
Both have another 2.
Both have a 3.
Each shared prime goes into both numbers.
Together they make the biggest common factor.
We multiply them.
GCF(24, 36) = 2 × 2 × 3
2 times 2 is 4.
4 times 3 is 12.
2 × 2 = 4
4 × 3 = 12
So the GCF is 12.
= 12
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: 12 goes into each number.
24 ÷ 12 = 2
36 ÷ 12 = 3
Nothing is left over.
So 12 is right!
Let's look back.
First: find the prime factors.
Then: multiply the shared primes.
Last: check.
