could anyone tell me if my resolution is correct?
Problem: Let $a$, $b$, and $c$ be integers satisfying:
- $a|(b+4c)$ $\qquad$ $(1)$
- $a|(b-2c)$ $\qquad$ $(2)$
- $a$ leaves remainder $1$ when divided by $3$ $\qquad$ $(3)$
Under these conditions can we say that $a$ divides $b$?
My solution:
Theorems and properties used:
$1$: If $a|b$ and $a|c$ then $a|(b+c)$;
$2$: If $a|b$ then $a|bc$ for any integer $c$;
$3$: If $a$ and $b$ are integers and $a = q*b + r$ where $q$ and $r$ are integers, then: $GCD(a,b)=GCD(b,r)$;
$4$: let $a$, $b$ and $c$ be three integers such that $a$ divides $bc$ and $a$ and $b$ are prime to each other, then $a$ divides $c$.
Using Theorem $2$ and relation $(2)$ we can say that:
$a|2(b-2c)$ $\longrightarrow$ $a|(2b-4c)$ $\qquad$ $(4)$
Using Theorem $1$ and relation $(4)$ we obtain that:
$a|[(b+4c)+(2b-4c)]$ $\longrightarrow$ $a|3b$ $\qquad$ $(5)$
Due to condition $(3)$ we can write $a$ as:
$|a|=3k+1$, $\qquad$ $k\in \mathbb{Z}$ $\qquad$ $(6)$
Using property $3$:
$GCD(a,3)=GCD(3,1)=1$ $\qquad$ $(7)$
Finally, due to the result obtained in $(7)$ and property $4$ we can conclude that $a|b$
I believe it's all right, but I still don't have much confidence if the properties and theorems I've chosen can be applied here or if I've applied them correctly.
If you can confirm that everything is correct, or if it is wrong, tell me where I went wrong and how to proceed, I would be grateful.