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**SlipEternal** The answer in post #1 is not correct. You are not trying to prove that if $\displaystyle x$ and $\displaystyle y$ are odd, then their product is odd. You are trying to prove that if $\displaystyle x\cdot y$ is odd, then both $\displaystyle x$ and $\displaystyle y$ are odd. So, if you assume that $\displaystyle x$ and $\displaystyle y$ are odd, you will always get that $\displaystyle x$ and $\displaystyle y$ are odd because that was your assumption in the first place. Instead, assume only that the product $\displaystyle x\cdot y$ 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 $\displaystyle x$ is even. Then $\displaystyle x=2k$ for some integer $\displaystyle k$. Hence $\displaystyle x\cdot y = 2ky = 2(ky)$ is even. This proves the contrapositive, and in so doing, also proves the initial statement.