Factoring Integers 

For example, a nontrivial factor of 11111 is 41, whereas trivial factors are 1, 1, 11111 and 11111.
If n is prime then it has no nontrivial factors. There are techniques for identifying nonprimes that do not explicitly exhibit a factor, so the question of finding a factor is interesting.
The RSA public key cryptosystem uses numbers that are hard to factor, and if a way could be found to factor numbers quickly then that would effectively break it.