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I have the following problem:

Let $c$ and $d$ be integers and let $C = \{ x \in \mathbb{Z} : x | c\}$ and $D = \{ x \in \mathbb{Z} : x | d \}$. Prove $C \subseteq D$ if and only if $c | d$

I had no problem prooving the second part of this statement ($c | d \Rightarrow C \subseteq D$), but I am stuck for the first part. I already have the following:

Assume $C \subseteq D$.

Let $x \in C$. By definition of $C$, $x | c$. By definition of divides, there exists an integer $a$ such that $c = xa$. Further we know, by definition of "is a subset of", that $x \in D$. By definition of $D$, $x | d$. By definition of divides, there exists an integer $b$ such that $d = xb$.

I then get stuck after a few more steps on the following:

$d = \displaystyle\frac{b}{a} c$

How would I prove that $\displaystyle\frac{b}{a}$ is an integer?

  • 2
    Hint: $c\in C$ and $C\subseteq D$. – 2017-02-23
  • 1
    Thank you so much for that hint, I got it now! – 2017-02-23

1 Answers 1

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Hint: one of the divisors of $c$ is $c$.