If a|b and b|c , (how can I prove) ⇒ a|c?

Question

I understand that if a|b, then amodb=0, thus a = qb (remainder is zero) and b = qc. How can I prove using this that a|c?

Answer 1

Note: From now, we will be assuming $j,k\neq 0.$

Since $a|b,$ we can say $b=ja$ for $j\in \mathbb{Z}.$ Similarly, $b|c$ so $c=bk$ for $k\in \mathbb{Z}.$

Substituting for $b$ yields $c=ajk.$ Clearly, $jk\in \mathbb{Z}$ so, our assertion is true. $\blacksquare$

Answer 2

By assumption, $na=b$ and $mb=c$ for some integers $m$ and $n$. Therefore $mna=c$ so that $c$ is an integer multiple of $a$ ( and of course $a|c$ by definition). $\blacksquare$