Question

Prove that for all integers a, b, c such that c =/= 0, if ac|bc then a|b.

Answer 1

ANSWER on Question #85799 – Math – Discrete Mathematics

QUESTION

Prove that for all integers $a, b, c$ such that $c \neq 0$, if $ac|bc$ then $a|b$.

SOLUTION

Notation $ac|bc$ means that there exists an integer $k$ such that

$$k \cdot (ac) = bc \rightarrow c(ka) = cb$$

We can divide the last equality $c(ka) = cb$ by $c$, since $c \neq 0$ by the condition.

Then,

$$c(ka) = cb \mid \div (c) \rightarrow ka = b \rightarrow a \mid b$$

Q.E.D.

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