Problem Solving

If x is an integer, and k = (x -1) (x +2) - (x -1) (x -2), then which of the following must be true?
(A) k is odd only when x is odd.
(B) k is odd only when x is even.
(C) k is even only when x is even.
(D) k is even only when x is odd.
(E) None of the above.



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Hi!

k = (x -1) (x +2) - (x -1) (x -2) = x^2 + x - 2 -(x^2 - 3x + 2) = 4x - 4 = 4(x - 1)
k = 4(x - 1)

(A)
if x is ODD then k = 4(ODD - 1) = 4*EVEN = EVEN*EVEN = EVEN
This answer choice is incorrect
(B)
if x is EVEN then k = 4(EVEN - 1) = 4*ODD = EVEN*ODD = EVEN
This answer choice is incorrect
(C)
if x is EVEN then k = 4(EVEN - 1) = 4*ODD = EVEN*ODD = EVEN
if x is ODD then k = 4(ODD - 1) = 4*EVEN = EVEN*EVEN = EVEN
This answer choice is incorrect
(D)
if x is ODD then k = 4(ODD - 1) = 4*EVEN = EVEN*EVEN = EVEN
if x is EVEN then k = 4(EVEN - 1) = 4*ODD = EVEN*ODD = EVEN
This answer choice is incorrect
(E)
only answer choice E is correct

KEY POINT:
ODD*ODD = ODD
EVEN*ODD = EVEN
ODD*EVEN = EVEN
EVEN*EVEN = EVEN

Hope it helps!
Best,
Maciek

IMO (E)

X = odd or even so lets say that x=2,3

k = (2 -1) (2 +2) - (2 -1) (2 -2)
--> k = 4
k = (3 -1) (3 +2) - (3 -1) (3 -2)
--> k = 8

so regardless of whether x = odd or even, k = even.

Hence (E).

One more for E. I plugged in 3 diff values and got diff answers each time. Hence, E.