I Have A Counterexample

Number Theory Level 1

It is given that the product of any two even numbers is even. Analogously, can we say that the product of any two odd numbers is odd as well?

Yes No

2 solutions

Aditya Kumar
Apr 25, 2016

Any odd number can be written in the form $2n+1$ , for any positive integral value of $n$ .

So we take two odd numbers $2a+1$ and $2b+1$ .

Therefore, the product is $4ab+2a+2b+1$ . Now, this can be written of the form: $2(2ab+a+b)+1$ . Clearly, this is of the form: $2n+1$ .

Hence, we can conclude that product of any two odd numbers is always odd.

Challenge Master Note: Simple standard approach. For the sake of variety, can you solve this question by proof by contraposition?

Pi Han Goh - 5 years, 1 month ago

Yes. This is just an example of biconditional operation.

Aditya Kumar - 5 years, 1 month ago

John Stotko
Apr 29, 2016

For a number to odd it cannot be divisable by two; it cannot have 2 as a factor. If a and b are factors of a number c, and two isn't a factor of a or b (that is they are odd), c cannot have a factor of 2.

I Have A Counterexample

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