Having trouble figuring out the issue with the question in my textbook
Here is a sample proof that contains an error. Explain why the proof is not correct.
Theorem: If a\b and b\c, then a\c.
Proof: Since a|b, b = a*k.
Since b|c, c = b*k.
Therefore c = bk = (ak)k = ak*k = a(k^2)
Therefore a|c
From my understanding (ak)k = ak^2 checks out and properties of divisibility rules state that a|b means that b is some multiple of a. Is the issue with the proof that there is no "assumption" being made at the start or some kind of notation error?
No. The error lies in the fact that the number $k$ such that $b=a\times k$ doesn't have to be such that $c=b\times k$. From the fact that $b\mid c$, what follows is that $c=b\times k'$ for some integer $k'$ (which doesn't have to be $k$). Therefore $c=a\times(k\times k')$. It follows that $a\mid c$.