How do i prove this supposition by contradiction

How can i prove the following supposition by contradiction and by contraposition?

For all in integers a, b, and c, if a divides b, and a doesn't divide c, then a doesn't divide (b+c)

Suppose a divides b, a doesn't divide c, and a divides (b+c).

Then a divides (b+c) - b = c.

This is a contradiction, so a doesn't divide (b+c).

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Oddly enough, the above is a direct proof. The proof by contradiction would instead have as an assumption
"it's false that a doesn't divide (b+c)"
and then it would have to imply
"a divides (b+c)".
Anyway, that's just a bit of logical nonsense.