Prove that if m and n are any two odd (integers) then mn is also odd.

Question

Here is what i have so far

By definition an integer is called odd if there exist an integer k such that n=2k+1 so if n and m are any two odd integers the product of those two integers is odd.

I dont know is this is the correct way of proving that mn is odd.

A start: For some integer $k$, $m=2k+1$. For some integer $l$, $n=2l+1$. Multiply. — André Nicolas, Mar 10 '16 at 02:41
Just spell out the details. Actually do the multiplication and put it in the $2k+1$ form, i.e. find which $k$ you need. — Derek Elkins left SE, Mar 10 '16 at 02:42
You recalled a definition and repeated the question but you didn't prove anything. Your "so if..." needs some argument. — Anne Bauval, Dec 11 '23 at 16:53

score 1 · Answer 1 · answered Mar 10 '16 at 03:38

1

Let $m$ and $n$ be two odd numbers. By the definition of odd, there exists integers $k$ and $l$ such that $m=2k+1$ and $n=2l+1$. We see that $mn=(2k+1)(2l+1)=(4kl+2k+2l)+1$. By the definition of even, $4kl+2k+2l$ is an even integer, so $mn$ is odd.

answered Mar 10 '16 at 03:38

Cooper Young

94

According to the definition you recalled, your last sentence is not the right way to prove that $mn$ is odd. Instead, write $$mn=2(2kl+k+l)+1.$$ – Anne Bauval Dec 11 '23 at 16:55

Prove that if m and n are any two odd (integers) then mn is also odd.

1 Answers1

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