If $a_1, a_2, \cdots, a_n$ are two-to-two relative prime integers (n ≥ 2) and for each i, i =1, 2, . ., n, $a_i|c$, prove that $a_1*a_2 · · · a_n|c$

Question

I did this proof by induction:

Case Base: n=2. if we have $a_1|c$ , $a_2|c$ and $(a_1,a_2)=1$ then $a_1$*$a_2$|c.

Inductive step:

Let's prove that $a_1$$a_2$...*$a_(n+1)$|$c$

I don't know how to use the hypothesis and I don't know how to finish this problem by induction.

However my attempt was this one:

We have ($a_1$$a_2$...$a_n$)|$c$. or $a_(n+1)$|$c$.
If $a_(n+1)$|$c$ we are done.
if ($a_1$$a_2$...$a_n$)|$c$ then $a_1$|$c$ for $n=1$,$2$,$3$,...,$n$. Then as hypothesis $a_1$$a_2$...*$a_n$ are two-to-two relative prime integers then $c=a_n$ for some $i=1,2,3,...,n$.

Someone can please help me out.