# proof by contrapositive help

• Oct 16th 2012, 02:24 PM
Tweety
proof by contrapositive help
Let a be an integer. Prove by contraposition that if a2 - 11a + 22 is odd then a is even.
Fill in the gaps in the following proof.

Suppose a is odd, so a = 2n + 1 for some integer n.
Then in terms of n, a2 - 11a + 22 = 2(..........)

so a2 - 11a + 22 is an(.........)

Hence, by contraposition, if
a2 - 11a + 22 is odd then a is even.

• Oct 18th 2012, 06:04 AM
emakarov
Re: proof by contrapositive help
Can you substitute a = 2n + 1 into a² - 11a + 22?
• Oct 18th 2012, 08:18 AM
Tweety
Re: proof by contrapositive help
thats what I was thinking, but am gona ask my teacher who wrote this, cause I dont understand what to do

and what is 2 being factored out of ?
• Oct 18th 2012, 09:11 AM
emakarov
Re: proof by contrapositive help
Quote:

Originally Posted by Tweety
and what is 2 being factored out of ?

2 is being factored out of the result of substitution of 2n + 1 for a in a² - 11a + 22. To do the substitution, replace a in a² - 11a + 22 with 2n + 1 and perform the simplification.