Question

Use a direct proof to show that the product of two odd integers is odd

Answer 1

Proved

Explanation:

Let a, b be two odd numbers

We know odd numbers give remainder as 1 when divided by 2

Hence a and b can be written as

`a=2m+1\\b=2n+1`, for some integers m and n.

Now product

=
`ab =(2m+1)(2n+1)\\= 4mn+2m+2n+1\\=2(2mn+m+n)+1`

We have when m and n are integers, 2mn+m+n also will be an integer say s

Then product
`ab =2s+1`

again gives remainder 1 when divided by 2

Hence product is odd.