For all integers a, b, c, if a | b and b | c then a | c. [duplicate]

Solution 1:

HINT: This is just a matter of applying the definitions, which is always the first thing that you should try. What does $a\mid b$ mean? It means that there is an integer $m$ such that $b=am$. Similarly, $b\mid c$ means that there is an integer $n$ such that $c=bn$. You now have the equations

$$\left\{\begin{align*} &b=am\\ &c=bn\;, \end{align*}\right.\tag{1}$$

and you want to show that $a\mid c$. Once again you should turn to the definition: this means that there is an integer $k$ such that $c=ak$. Can you combine the equations in $(1)$ to find such an integer $k$?

Solution 2:

$$ a \mid b \Rightarrow b= na , n \in \Bbb Z$$

$$ b \mid c \Rightarrow c = kb = (kn)a \Rightarrow a \mid c , k \in \Bbb Z $$