# Conjectures About Integers

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Conjectures about integers test if you are familiar with properties of integers and their classifications. Review the types of integers if you do not know the definition of even, odd, positive, negative, consecutive integers, prime numbers, etc.

You will need to be able to verify solutions, check cases, find counterexamples and consider multiple scenarios.

## Odd and Even Integers

## If \(x\) and \(y\) are both integers, which of the following must be even? \[\]

\(\quad \text{A) } x + y \)

\(\quad \text{B) } 2x + y \)

\(\quad \text{C) } x^2 + y^2 \)

\(\quad \text{D) } (x+y)^2 - ( x - y)^ 2 \)

\(\quad \text{E) } x^2 + xy + y^2 \)

A) If \( x = 1, y = 2, \) then \( x + y = 3, \) which is not even.

B) If \( x = 2, y =1, \) then \( 2x+y =5, \) which is not even.

C) If \( x = 1, y =2, \) then \( x^2 + y^2 = 5, \) which is not even.

D) \( (x+y)^2 - (x-y)^2 = ( x^2 + 2xy + y^2) - ( x^2 - 2xy + y^2) = 4xy \). Since we're multiplying by 4, this must be even.

E) If \( x = 1, y = 2, \) then \( x^2 + xy + y^2 = 1 + 2 + 4 = 7, \) which is not even.Hence the answer is D.

**Cite as:**Conjectures About Integers.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/conjectures-about-integers/