I tried to disprove the statement below. Are my steps valid?
Let a, b and c be positive integers.
Disprove: $$c \nmid b \implies c \nmid (2a + 7b) \lor c \nmid(a - b)$$
Goal: Provide a case where the premise is true but the conclusion is false.
My Proof:
Consider when a=4, b=1, and c=3.
Then $$3 \nmid 1$$ is true, and $$3 \nmid 15 \lor 3 \nmid 3 $$ is false. Showing that there is a case where assuming $$c \nmid b $$ leads to a false conclusion.
If my steps are valid, how can I improve my wording / organization of the proof?
Is there a better way to come up with values for a, b, and c other than brute-force?