Classical intro to modern Number theory I'm self-studying Classical Intro to Modern Number Theory, by Kenneth Ireland and Michael Rosen, and I am stuck on a simple proof on page $34$: 

Suppose $a_1, a_2 ...,a_t$ all divide $n$, and that $gcd(a_i, a_j) = 1$ for $i \neq j$. then $a_1\cdot\ a_2\cdot \ldots \cdot a_t$ all divide $n$. 

The book proves by induction: 

$a_1\cdot a_2\cdot \dots a_{t-1}$ divide $n$. Then $gcd(a_t, a_1\cdot \ldots a_{t-1}) = 1$. Then $\exists$ $r, s$ such that $r\cdot a_t + s\cdot a_1\cdot \ldots a_t = 1$. Multiply both sides by $n$. Inspection shows that the left-hand side is divisible by $a_1\cdot a_2\ldots \cdot a_t$ and the result follows.

I don't understand the multiply by $n$ and inspection part. It seems straight forward but I'm spacing. 

Isn't it much easier just using prime factor decomposition, writing $n=\prod p_i^{k_i}$ and noticing that by definition, as $gcd(a_i,a_j)=1$ and $a_i\mid n$ each $a_i$ is a subproduct of the $p_i's$ that make up $n$ and all subproducts use distinct sublist of $p_i's$? 
