If x and y are positive integers, is the expression x^x + y^y + x^y ev

Question

If x and y are positive integers, is the expression x^x + y^y + x^y even?
(1) y^x + x^y is even.
(2) (x+3)(2y + 1)is odd.

gmatophobia · Answer

Question: Is the expression  x^x + y^y + x^y  even

Statement 1

(1)  y^x + x^y  is even.

This statement holds true under two conditions 
  both x and y are even 
 both x and y are odd

Case 1: both x and y are even

x^x + y^y + x^y  = even + even + even = even

Case 2: both x and y are odd

x^x + y^y + x^y  = odd + odd + odd = odd

As we have two contradicting results, the statement alone is not sufficient to answer the question.

We can eliminate A and D.

Statement 2

(2)  (x+3)(2y + 1)  is odd.

The product of two integers is odd when both integers are odd.

Hence, 
(x+3) = odd → x = even
(2y + 1) = odd → 2y = even
y can be even or y can be odd.

Case 1: x = even and y = even

x^x + y^y + x^y  = even + even+ even = even

Case 2: x = even and y = odd

x^x + y^y + x^y  = even + odd + even = odd

As we have two contradicting results, the statement alone is not sufficient to answer the question. We can eliminate B.

Combined

From statement 1, x and y have the same even-odd nature. Hence, both x and y are even.

x^x + y^y + x^y  = even + even + even = even

We now have a definite answer.

Option C

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