Prove the following for all integers x and y if x is odd and

Prove the following: for all integers x and y, if x is odd and y is odd, then x middot y is odd.

Solution

Let x and y be two integers such that both are odd integers.

then, they can be written as

x = 2n + 1

y = 2m + 1

where n and m are integers.

Now

x.y = ( 2n + 1 ) ( 2m + 1) = 2n ( 2m + 1 ) + 1( 2m + 1)

= 4mn + 2n + 2m + 1

=2 ( 2mn + n + m ) + 1

since m and n are integers, so 2mn, m + n are also integers and 2mn + n + m is an integer.

Let t = 2mn + m + n

then

x.y = 2t + 1

where t is an integer.

Hence, x.y is odd.