Question

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

Answer 1

Answer:

Proved

Step-by-step 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.