How to tell if two numbers are coprime

Coprime is a word about a pair, never about one number on its own. 9 is not coprime, but 9 and 10 are.

The rule

Two numbers are coprime when the only whole number that divides both of them is 1.

How to check it

Write down every number that divides the first one.
Do the same for the second one.
Compare the two lists. 1 is always in both.
If 1 is the only number in both lists, the pair is coprime.

Worked examples

Are 3 and 4 coprime?

3 divides by 1 and 3
4 divides by 1, 2 and 4

Only 1 is in both lists, so yes.

Are 2 and 6 coprime?

2 divides by 1 and 2
6 divides by 1, 2, 3 and 6

2 is in both lists as well, so no.

Are 9 and 10 coprime?

9 divides by 1, 3 and 9
10 divides by 1, 2, 5 and 10

Yes, even though neither number is prime.

Are 8 and 27 coprime?

8 divides by 1, 2, 4 and 8
27 divides by 1, 3, 9 and 27

Only 1 is in both lists, so yes. 8 is made of 2s and 27 of 3s, so they can never share a factor.

Why divisibility tests care

Two tests together only prove divisibility by the numbers multiplied when the pair is coprime. 6 is 2 × 3, and 2 and 3 are coprime, so passing both tests proves a number divides by 6.

2 and 6 are not coprime, because 2 divides both. That is why testing 2 and 6 does not prove a number divides by 12: 18 passes both and is not a multiple of 12. Testing 3 and 4 does prove it, because 3 and 4 are coprime.

Where mistakes happen

Reading it as another word for prime. 9 and 10 are coprime and neither of them is prime.
Expecting the pair to share nothing at all. Every pair shares 1, and the rule is that 1 is the only thing they share.

Questions people ask

Is 1 coprime with every number?

Yes. The only number that divides 1 is 1 itself, so 1 is the only thing any pair containing it can share.

Are two different prime numbers always coprime?

Yes. A prime divides by 1 and itself, so two different primes have nothing in common but 1. Two copies of the same prime are not coprime, because that prime divides both.

Is there another name for it?

You will also see relatively prime and mutually prime. All three mean the same thing.

Are 5 and 10 coprime?

No. 5 divides both of them, because 10 = 2 × 5.

Are 4 and 8 coprime?

No. 2 and 4 both divide 4 and 8. When one number divides the other, the pair is never coprime, unless the smaller one is 1.

How do I check a big pair quickly?

Look for a number that divides both, rather than listing everything. If you find one that is not 1, you are done and the answer is no.

Is there a name for the biggest number a pair shares?

Yes, the highest common factor. Saying a pair is coprime is the same as saying their highest common factor is 1.