If x and y are positive integers, is x a multiple of y?

Question

(1) y^2 + y is not a factor of x
(2) x^3 – x is not a multiple of y

gmatophobia · Answer

Question  : Is  \frac{x}{y}  an integer

Statement 1

Y^2 + Y is not a factor of X

Inference  :  \frac{y(y+1)}{x}  is not an integer

However, using this inference we cannot comment whether  \frac{x}{y}  is an integer

Case 1: x = 4, y = 2

\frac{3(2)}{4}  is not an integer; however  \frac{4}{2}  is an integer

Case 2: x = 5, y = 6

\frac{6(7)}{5}  is not an integer, and  \frac{5}{6}  is also not an integer

Hence, statement 1 is alone not sufficient. We can eliminate A and D.

Statement 2

X^3 – X is not a multiple of Y

Inference  :  \frac{x(x+1)(x-1)}{y}  is not an integer

Therefore y doesn't completely divide either x or (x+1) or (x-1) or the product of the three numbers.

Hence, we can be sure that  \frac{x}{y}  is NOT an integer.

Statement 2 is alone sufficient.

Option B

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