Why if, a/b and b/c, a/c ? b|a . c|b -> c|a

Question

I'm reading An Introduction to the Theory of Numbers By Godfrey Harold Hardy, G. H. Hardy, E. M. Wright, Roger Heath-Brown, Joseph Silverman.

I've noticed that the transitive property (if a=b and b=c, a=c) is applied. I feel I'm missing something. If you have any suggestions to read something before this or a piece that focuses on this subject, even extremely in depth, that'd be greatly appreciated!

Answer 1

This is just a direct application of the definition of divisibility. If $a \mid b$, this means (by definition) that $b=ak$ for some integer $k$. If $b \mid c$ this likewise means $c=bm$ for some $m$. Hence $$ c=bm=akm $$ and so $a \mid c$.

I never used transitivity to prove this. The claim itself is that divisibility is transitive.