Can b be an integer if it does not divide k for every natural number k?

[pjgrah01]

Homework Statement

Prove by contradiction that if b is an integer such that b does not divide k for every natural number k, then b=0.

Homework Equations

The Attempt at a Solution

I know that proof by contradiction begins by assuming the false statement: If b is an integer such that b does not divide k for every kεℕ, then b≠0, which is equivalent to "there exists an integer b such that b does not divide k and b≠0, for every kεℕ. But I'm not sure how to proceed from here.

 

[mfb]

I know that proof by contradiction begins by assuming the false statement: If b is an integer such that b does not divide k for every kεℕ, then b≠0

That is not the opposite statement.
"If a pen is green, then it is my pen" is wrong, but "if a pen is green, then it is not my pen" is also wrong (because I own some green pens, but not all).

 

[HallsofIvy]

Also "b does not divide k for every natural number k" is itself ambiguous. It could be read as "b does not divide any natural number" but here, I think, is intended to say "there exist a natural number, k, that b does not divide". The "contradiction" would just be "there exist b, not equal to 0, such that b does not divide any natural number, k." Given that, what can you say about k= 2b?