Is x divisible by 3?

[BTGmoderatorAT]

Is x divisible by 3?

(1) x + y is divisible by 3.
(2) x - y is divisible by 3.

What's the best way to determine which statement is sufficient? Can any experts help?

[Jay@ManhattanReview]

Is x divisible by 3?

(1) x + y is divisible by 3.
(2) x - y is divisible by 3.

What's the best way to determine which statement is sufficient? Can any experts help?

This question can be dealt with an ease by choosing smart numbers for x and y.

(1) x + y is divisible by 3.

Case 1: Say x = 3 and y =0, then x + y = 3 + 0 = 3. We see that x + y and x are divisible by 3. The answer is Yes.
Case 2: Say x = 4 and y = 2, then x + y = 5 + 2 = 6. We see that though x + y is divisible by 3, x is NOT. The answer is No.

No unique answer. Not sufficient.

(2) x - y is divisible by 3.

Case 1: Say x = 3 and y =0, then x - y = 3 - 0 = 3. We see that x - y and x are divisible by 3. The answer is Yes.
Case 2: Say x = 5 and y = 2, then x - y = 5 - 2 = 3. We see that though x - y is divisible by 3, x is NOT. The answer is No.

No unique answer. Not sufficient.

(1) and (2) together

You cannot find a pair of integers such that x + y and x - y are each divisible by 3, and x is not. Thus, x is divisible by 3. The answer is Yes. Sufficient.

Let's take an algebraic route to understand this.

Say x + y = 3k, and x - y = 3q, where k and q are any integers

Thus, x = 3[(k + q)/2]; and y = 3[(k - q)/2]

We see that x is a multiple of 3. Sufficient.

The correct answer: C

Hope this helps!
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[Brent@GMATPrepNow]

Is x divisible by 3?

(1) x + y is divisible by 3.
(2) x - y is divisible by 3.

What's the best way to determine which statement is sufficient? Can any experts help?

This question can be dealt with an ease by choosing smart numbers for x and y.

(1) x + y is divisible by 3.

Case 1: Say x = 3 and y =0, then x + y = 3 + 0 = 3. We see that x + y and x are divisible by 3. The answer is Yes.
Case 2: Say x = 4 and y = 2, then x + y = 5 + 2 = 6. We see that though x + y is divisible by 3, x is NOT. The answer is No.

No unique answer. Not sufficient.

(2) x - y is divisible by 3.

Case 1: Say x = 3 and y =0, then x - y = 3 - 0 = 3. We see that x - y and x are divisible by 3. The answer is Yes.
Case 2: Say x = 5 and y = 2, then x - y = 5 - 2 = 3. We see that though x - y is divisible by 3, x is NOT. The answer is No.

No unique answer. Not sufficient.

(1) and (2) together

You cannot find a pair of integers such that x + y and x - y are each divisible by 3, and x is not. Thus, x is divisible by 3. The answer is Yes. Sufficient.

Let's take an algebraic route to understand this.

Say x + y = 3k, and x - y = 3q, where k and q are any integers

Thus, x = 3[(k + q)/2]; and y = 3[(k - q)/2]

We see that x is a multiple of 3. Sufficient.

The correct answer: C

For the answer to be C, we need some sort of restriction that states x and y are integers.
Otherwise, non-integer pairs of values such as x = 4.5 and y = 1.5 satisfy both statements.

Cheers,
Brent

[Vincen]

Is x divisible by 3?

(1) x + y is divisible by 3.
(2) x - y is divisible by 3.

What's the best way to determine which statement is sufficient? Can any experts help?

Hello. This is how I'd solve it:

(1) x+y is divisible by 3.

If we take x=3 and y=6 we have that x+y=9 and 9 is divisible by 3. This could implie that x must be divisible by 3; but is we take x=1 and y=5 then x+y=6 and in this case x is not divisible by 3. Therefore, this statement is INSUFFICIENT.

(2) x-y is divisible by 3.

The same as above, take first x=6 and y=3, then take x=5 and y=2. INSUFFICIENT.

Now, using both statements together $$x+y=3\cdot k$$ $$x-y=3\cdot p$$ this implies that $$2x=3k+3p\ \Leftrightarrow\ \ \ x=\frac{3\left(k+p\right)}{2}$$ and this last number is divisible by 3 always.

Hence, this case is SUFFICIENT.

The correct option is C.