If x is an integer, is (x + p)(x + q) an even integer?

Analyzing the question:

Since even*any integer= even, We need to know whether x+p or x+q is even.

Statement 1:

x + p can be either even or odd. Same thing for x+q. Insufficient.

Statement 2:

Same as in (1). Insufficient.

Combined:

If x is even, x + p and x + q are both even, making the product even. If x is odd, x + p and x + q are both odd, making the product odd. Therefore insufficient.

Ans: E

Given: X = An integer
To find: (x + p)(x + q) an even integer?

Solution:
(1) q is an even integer.
Let Q = 2
Case 1: X & P is odd = 3
=> (x + p)(x + q)
=> (3 + 3)(3 + 2)
=> (6) (5)
=> 30 (even, Yes)

Case 2: X is odd = 3 & P is even = 4
=> (x + p)(x + q)
=> (3 + 4)(3 + 2)
=> (7) (5)
=> 35 (even, NO)
(Insufficient)

(2) p is an even integer.
Let P = 2
Case 1: X is odd = 3 & Q is even = 4
=> (x + p)(x + q)
=> (3 + 2)(3 + 4)
=> (5) (7)
=> 35 (even, NO)

Case 2: X & Q is even = 4
=> (x + p)(x + q)
=> (4 + 2)(4 + 4)
=> (6) (8)
=> 48 (even, Yes)
(Insufficient)

Combining both:
P = 2 and Q = 4
Case 1: X = 3 (odd)
=> (x + p)(x + q)
=> (3 + 2)(3 + 4)
=> (5) (7)
=> 35 (even, No)

Case 1: X = 4 (even)
=> (x + p)(x + q)
=> (4 + 2)(4 + 4)
=> (6) (8)
=> 48 (even, Yes)
(Insufficient)

Answer is E

Posted from my mobile device

We need either (x + p) or (x + q) to be even for the product to be even.

Statement 1:

(x + q) is not always even and we don't know the parity of (x + p), insufficient.

Statement 2:

(x + p) is not always even and we don't know the parity of (x + q), insufficient.

Combined:

If x is even we get Even*Even = Even. If x is odd, we get Odd*Odd = Odd. Insufficient.

Ans: E

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