I am given the following proposition:
Let $A_1,A_2,...,A_s$ be odd integers with $A_i > 1$ for $i$, $1 \leq i \leq s$. Prove: if $n = A_1A_2...A_s+2$, then $n$ is not divisible by $A_i$, for every $i, 1 \leq i\leq s$.
I have identified that I should be using strong induction to solve this problem but am stuck on where to actually start.
You don't need to use induction here. Suppose $A_{1},\ldots, A_{s}$ are odd numbers greater than $1$. Let $n=A_{1}\cdots A_{s}+2$. We need to show that $n$ is not divisible by any of the $A_{i}$'s. Suppose on contrary that for some $1\leq i\leq s$, $A_{i}$ divides $n$. Clearly $A_{i}$ also divides $A_{1}\cdots A_{s}$. Thus, $A_{i}$ divides the difference $n-A_{1}\cdots A_{s}=2$. But this is impossible since $A_{i}\geq 3$.