If x is an integer, is y divisible by 3? (1) y = 2x^3 + 9x^2 - 5x

Question

If x is an integer, is y divisible by 3?

(1) y = 2x^3 + 9x^2 - 5x
(2) x is not divisible by 3

gmatophobia · Answer

Statement 1

(1) y = 2x^3 + 9x^2 - 5x

y = x(2x^2 + 9x - 5)

y = x(2x^2 + 10x - x - 5)

y = x(2x(x+5)-1(x+5))

y = x(x+5)(2x-1)

Case 1: x is a multiple of 3

If x is a multiple of 3, the product  x(x+5)(2x-1)  is a multiple of 3. Hence, y is a multiple of 3.

Case 2: x is not a multiple of 3

If x is not a multiple of 3, the remainder of  \frac{x}{3}  is either 1 or 2.

2(a)   : If the remainder of  \frac{x}{3}  = 1

(x+5) will be a multiple of 3.

Remainder  \frac{(x+5)}{3}  = Remainder ( \frac{x}{3} ) + Remainder ( \frac{5}{3} ) = 1 + 2 = 3

Remainder ( \frac{3}{3} ) = 0

2(b)   : If the remainder of  \frac{x}{3}  = 2

(2x-1) will be a multiple of 3.

Remainder  \frac{(2x-1)}{3}  = Remainder ( \frac{2x}{3} ) - Remainder ( \frac{1}{3} ) = 2*2 -1 = 3

Remainder ( \frac{3}{3} ) = 0

Hence, in all cases, y is a multiple of 3. The statement alone is sufficient and we can eliminate B, C, and E.

Statement 2

(2) x is not divisible by 3

Clearly not sufficient as we do not know any relationship between x and y.

IMO Option A

Nabneet · Answer

St 1:
y = x(2x-1)(x+5)
 = x (2x^2 + 9x -5)
 = x (2x-1) (x+5)
Inserting any values of X, y is divisible by 3

Sufficient

St2:
No info about y.
Not Sufficient

Answer: A

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