x and y are positive integers. If xy + x is odd, then which of the following must be even?

A. x
B. y
C. x + y
D. xy − x
E. x^2 − y
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Some important rules regarding the products of evens and odds:
1. (ODD)(ODD) = ODD
2. (ODD)(EVEN) = EVEN
3. (EVEN)(EVEN) = EVEN
4. ODD +/- ODD = EVEN
5. ODD +/- EVEN = ODD
6. EVEN +/- EVEN = EVEN
As we can see from #1, IF the product of two integers is ODD, then the two integers must both be odd.

Given: xy + x is ODD
If we FACTOR xy + x, we get x(y + 1)
So, we're told that x(y + 1) is ODD
From rule #1 above, we can conclude that x is ODD and (y + 1) is ODD
If (y + 1) is odd, then we can be certain that y is EVEN

Now check the answer choices:
A. x = ODD
B. y = EVEN (done!!)
C. x + y = ODD + EVEN = ODD
D. xy − x = (ODD)(EVEN) - ODD = EVEN - ODD = ODD
E. x^2 − y = (ODD)(ODD) - EVEN = ODD - EVEN = ODD

Answer:

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Given that xy + x is odd, which means x(y+1) is odd.
Product of two nos. will be odd only if both the nos. are odd.
Hence x is odd, y+1 is also odd.
y+1 is odd means y is even. Answer will be B.

We are given that x and y are positive integers and that xy + x is odd, or x(y + 1) is odd, which means that x is odd and y is even. Thus, y MUST be even.

Answer: B
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