I have the idea of using the transitive property and/or the the integer combination property. I am stuck tho.

Question

\begin{equation} a, b, \text { and } c \text { are integers. Prove that if } a |(b-1) \text { and } 5 a |(c+2), \text { then } a |(2 b+c) \end{equation}

Answer 1

Note that if $5a|c+2$, then $a|c+2$ also (transitive property). Thus, because $a|b,c \implies a|b+c$ (integer combination property), we know $a|2(b-1)+(c-2)=2b+c$ as required.

Edit added OP terminology to proof for clarity

Answer 2

If $b-1=ka$ and $c+2=5ma$ then: $$2b+c=2(ka+1)+(5ma-2)=(2k+5m)a$$