If x, y, and d are integers and d is odd, are both x and y divisible by d ?

[BTGmoderatorDC]

If x, y, and d are integers and d is odd, are both x and y divisible by d ?

(1) x + y is divisible by d.
(2) x − y is divisible by d.



OA C

Source: Official Guide

[Jay@ManhattanReview]

If x, y, and d are integers and d is odd, are both x and y divisible by d ?

(1) x + y is divisible by d.
(2) x − y is divisible by d.

OA C

Source: Official Guide

Let's take each statement one by one.

(1) x + y is divisible by d.

Case 1: Say x = 4; y = 2 and d= 3; thus, we see that neither x nor y is divisible by d. The answer is no.
Case 2: Say x = 3; y = 0 and d= 3; thus, we see that both x and y are divisible by d. The answer is yes.

No unique answer. Insufficient.

(2) x − y is divisible by d.

Case 1: Say x = 4; y = 1 and d= 3; thus, we see that neither x nor y is divisible by d. The answer is no.
Case 2: Say x = 3; y = 0 and d= 3; thus, we see that both x and y are divisible by d. The answer is yes.

No unique answer. Insufficient.

(1) and (2) together

Say x + y = pd and x – y = qd, where p and q are positive integers

Thus,

x =d(p + q)/2; we see that x is divisible by d; similarly,
y =d(p – q)/2; we see that x is divisible by d.

The correct answer: C

Hope this helps!

-Jay
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[Brent@GMATPrepNow]

If x, y, and d are integers and d is odd, are both x and y divisible by d ?

(1) x + y is divisible by d.
(2) x − y is divisible by d.



OA C

Source: Official Guide

Given: x, y, and d are integers and d is odd

Target question: Are both x and y divisible by d?

Statement 1: x+y is divisible by d.
Let's TEST some values.
There are several values of x, y and d that satisfy statement 1. Here are two:
Case a: x = 6, y = 9 and d = 3. Notice that 6+9 = 15, and 15 is divisible by 3. In this case, the answer to the target question is YES, x and y ARE both divisible by d
Case b: x = 2, y = 4 and d = 3. Notice that 2+4 = 6, and 6 is divisible by 3. In this case, the answer to the target question is NO, x and y are NOT both divisible by d
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x-y is divisible by d
Let's TEST some values.
Case a: x = 9, y = 3 and d = 3. Notice that 9-3 = 6, and 6 is divisible by 3. In this case, the answer to the target question is YES, x and y ARE both divisible by d
Case b: x = 10, y = 4 and d = 3. Notice that 10-4 = 6, and 6 is divisible by 3. In this case, the answer to the target question is NO, x and y are NOT both divisible by d
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Here are some nice divisibility rules:
Rule #1. If integers A and B are each divisible by integer k, then (A + B) is divisible by k
Rule #2. If integers A and B are each divisible by integer k, then (A - B) is divisible by k


Statement 1 tells us that x+y is divisible by d
Statement 2 tells us that x-y is divisible by d
By Rule #1, (x+y) + (x-y) is divisible by d
Simplify to get: 2x is divisible by d
Since we're told that d is ODD, we know that x MUST BE divisible by d

By Rule #2, (x+y) - (x-y) is divisible by d
Simplify to get: 2y is divisible by d
Since we're told that d is ODD, we know that y MUST BE divisible by d

So, the answer to the target question is YES, x and y ARE both divisible by d

Answer: C

Cheers,
Brent