July 29, 2014

Homework Help: discrete math

Posted by romulo on Wednesday, March 7, 2007 at 1:45am.

Prove by contradiction that for any even integer a and any odd integer b, 4 does not divide (a^2 + 2b^2).

Proposition: That 4k (k is any integer) = a^2 +2b^2, and a is even, and b is odd.
But 4k is even (product of any integer and 4), so a^2 must be even, as 2b^2 is even.
Dividing both sides by 4,
k=a^2/4 + 2b^2/4
but a is even, so a=2*n where n is an integer. a^2=4n^2
k= n^2 + b^2/2

But b is odd, so b^2/2 is not an integer.

Therefore, k cannot be an integer, so the proposition is contradicted.

Answer this Question

First Name:
School Subject:

Related Questions

Discrete Math - 1. Assume that n is a positive integer. Use the proof by ...
discrete math - prove that if n is an integer and 3n+2 is even, then n is even ...
DISCRETE MATHS - We need to show that 4 divides 1-n2 whenever n is an odd ...
Algebra - The sum of two consecutive odd integers is 56. A. Define a variable ...
maths - prove that any odd positive integer of 8q+1,where q is any integer?
Algebra - The sum of two consecutive even integers is 118. A. Define a variable ...
DISCRETE MATHS - Prove that if n is an odd positive integer, then 1 ≡ n2 (...
Discrete Mathematics - Prove that if n is an odd positive integer, then 1 ≡...
math - if the number represented by n-3 is an odd integer, which expression ...
algebra - Kayla Wants To write an Expression that will always produce an odd ...