In the Book of Proof (Richard Hammack) there's an exercise that asks you to prove this:
Suppose $a$ is an integer. If $7|4a$ then $7|a$.
My attempted solution was simply this:
If $7|4a$ then there is an integer $b$ such that
$4a=7b$
Then $a=\frac{7b}{4}$
which is obviously divisible by 7. If it's necessary to be extremely explicit I suppose I could add
Let $c=\frac{b}{4}$. Then $a=7c$, and so $7|a$, from the definition of divisibility.
But when I looked at the solution provided in the back of the book, it was way more complicated than this, and involved the definitions of odd and even numbers, and uses a few more dummy variables.
I can follow the solution, but what I don't understand is, what's wrong with my solution?
The problem is that you are implicitly assuming that $\frac b4$ is an integer, which is actually what you're trying to prove. If you find yourself writing fractions in these kinds of problems, you're usually doing something wrong.