The answer in post #1 is not correct. You are not trying to prove that if

and

are odd, then their product is odd. You are trying to prove that if

is odd, then both

and

are odd. So, if you assume that

and

are odd, you will always get that

and

are odd because that was your assumption in the first place. Instead, assume only that the product

is odd. A logically equivalent statement to the one you are trying to prove would be "If x is even or y is even, then x*y is even." This is called the contrapositive, and a conditional statement is logically equivalent with its contrapositive. So, if you prove that statement, you also prove the original statement. Assume (without loss of generality) that

is even. Then

for some integer

. Hence

is even. This proves the contrapositive, and in so doing, also proves the initial statement.