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?

  • 3
    $\begingroup$ 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 $k\ge 1$ $\endgroup$ – Peter Feb 23 at 13:13
  • $\begingroup$ 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. $\endgroup$ – 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 $b=ak>0$.


Here is a direct proof:

$a,b > 0 \implies k = \dfrac{b}{a} > 0 \implies k \ge 1$, since $k$ is an integer.


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