I agree with your conclusions that "it is divisible by" some number. I don't think that you can conclude that something doesn't divide your number just because this test fails.

Concretely, your conclusion that 3|2^60-1 because of (x-1)|(x^n-1) is good.

I don't think you can conclude that 17 (does not divide) 2^60-1 because none of those factorizations yields 17.