# For integers a, b and k, if a and b are positive and b = ak, then k ≥ 1

Number theory: For integers $$a, b$$ and $$k$$, if $$a$$ and $$b$$ are positive and $$b = ak$$, then $$k \ge 1$$.

How would I proceed to answer this question? I was thinking of using contradiction but I don't know if it would lead me to the right answer?

• Contradiction is a way, but the easiest way is to consider that $k=\frac{b}{a}$ must be positive. Together with the assumption that $k$ is an integer, you can easily conclude $k\ge 1$ – Peter Feb 23 at 13:13
• The thing to realize is that with integers: $k \ge 1$ is equivalent to $k > 0$. So if you can prove $k> 0$ you are done. – fleablood Feb 26 at 1:01

If $$k<1$$, then $$k\leq 0$$ (because $$k\in\mathbb Z$$), so $$ak\leq 0$$ (because $$a\geq 1$$ and $$k\leq 0$$), which contradicts $$b=ak>0$$.
$$a,b > 0 \implies k = \dfrac{b}{a} > 0 \implies k \ge 1$$, since $$k$$ is an integer.