Testing Primality 

This doesn't work the other way, but this is a case where we know the answer to the "Yes/No" question, without getting the actual answer we want, a factor of n.
As a specific case, consider 6. is 31, not a multiple of 6. We claim therefore that there are divisors of 6. Consider 15. is 16383 which is not a multiple of 15, so 15 is not prime. But 17? is 65535, which is 17*3855.
The test can tell us a number is composite without providing a factor.
(It doesn't always work in the other direction, so be careful!)