discrete math

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.

  1. 👍 0
  2. 👎 0
  3. 👁 200
asked by romulo

Respond to this Question

First Name

Your Response

Similar Questions

  1. discrete math

    prove that if n is an integer and 3n+2 is even, then n is even using a)a proof by contraposition b)a proof by contradiction I'll try part b, you'll have to refresh me on what contraposition means here. Here is the claim we start

    asked by audryana on September 27, 2006
  2. Discrete Math

    1. Assume that n is a positive integer. Use the proof by contradiction method to prove: If 7n + 4 is an even integer then n is an even integer. 2. Prove: n is an even integer iff 7n + 4 is an even integer. (Note this is an if and

    asked by Math help on February 21, 2008
  3. Algebra 1 Polynomials

    Suppose n is an integer. Select all statements below that are true: (choose 3) A) n^2 + n is always an even integer*** B) n^2 + n is always an even integer when n is even*** C) n^2 + n is always an even integer when n is odd*** D)

    asked by Gerard Way on March 8, 2017
  4. Algebra

    The sum of two consecutive odd integers is 56. A. Define a variable for the smaller integer. B. What must you add to an odd integer to get the next greater odd integer? C. Write an expression for the second integer. D. Write and

    asked by Bri on October 14, 2011
  5. Math

    Let a, b, and m be integers, and m ≥ 2. Prove that: ab ≡ [ (a mod m) · (b mod m) ] (mod m). So I tried proof by cases: Assume ab ≡ [(a mod m) ∙ (b mod m)] mod m is true. Then ab mod m = [(a mod m) ∙ (b mod m)] mod m,

    asked by Kid on November 9, 2018

    We need to show that 4 divides 1-n2 whenever n is an odd positive integer. If n is an odd positive integer then by definition n = 2k+1 for some non negative integer, k. Now 1 - n2 = 1 - (2k+1)2 = -4k2-4k = 4 (-k2-4k). k is a

    asked by Don(please check my math) on October 24, 2013
  7. maths

    prove that any odd positive integer of 8q+1,where q is any integer?

    asked by sandra on April 17, 2012
  8. Algebra

    Find two consecutive odd integers such that 5 times the first integer is 12 more than 3 times the second. I've gotten this far but confused as to what the two intergers are. 5x=3(x+2)=12 5x=3x+2+12 5x=3x+14 2x=14 2x/2x=14/2 x=7

    asked by Amanda on April 22, 2007
  9. algebra

    Find the two consecutive odd integers such that 5 times the first integer is 12 more than 3 times the second. can some explain to me how to get this i was thinking the formula would be ' 5x+12=3x No, that's not it. The second

    asked by laura on March 29, 2007
  10. Algebra Word Problem

    Solve algebraically using one variable: Find three consecutive odd integers such that the product of the first integer and the third integer is equal to nine more than twelve times the middle integer.

    asked by Tony on February 20, 2017

More Similar Questions