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

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

A) \( x + y \).

B) \( 2x + y \).

C) \( x^2 + y^2 \).

D) \( (x+y)^2 - ( x - y)^ 2 \).

E) \( x^2 + xy + y^2 \).

Solution:
Let's consider A. If \( x = 1, y = 2 \) then \( x + y = 3 \) which is not even.

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

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

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

Let's consider 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.

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