Anaira Mitch wrote:If x and y are positive integers and x + y = 3^x, is y divisible by 6?

(1) x is odd.

(2) x is a multiple of 3.

We have to see whether y divisible by 6.

We have x + y = 3^x.

Let us take each statement one by one.

S1: x is odd.

@ x = 1

x + y = 3^x => 1 + y = 3^1 => y = 2. y is not divisible by 6. The answer is NO.

@ x = 3

x + y = 3^x => 3 + y = 3^3 => y = 27 - 3 =24. y is divisible by 6. The answer is YES. No unique answer. Insufficient.

S2: x is a multiple of 3.

@ x = 3

We already saw in statement 1 that @x=3, y is divisible by 6. The answer is YES.

@ x = 6

x + y = 3^x => 6 + y = 3^6 => y = 3^6 - 6 = Odd - Even = An odd number. y, an odd number is not divisible by 6, an even number. The answer is No. No unique answer. Insufficient.

S1 and S2:

From both the statements, we have x = an odd multiple of 3, i.e., x = 3/9/15/21/27, etc.

We have already seen that @x=3, y is divisible by 6. Let's test it @x=9

@ x = 9

x + y = 3^x => 9 + y = 3^9 => y = 3^9 - 3^2 = 3^2(3^7 - 1). Since 3^2 is a multiple of 3 and (3^7 - 1) = an even number, thus 3^2(3^7 - 1) = y must be divisible by 3*2 = 6. The answer is YES.

There is no need to analyze further. At higher values of x such as 15/21/27, etc, you would find that y = 3^x - x would always be an even multiple of 3, making it divisible by 6. Sufficient.

The correct answer: C

Hope this helps!

Relevant book: Manhattan Review GMAT Data Sufficiency Guide

-Jay
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Anaira Mitch wrote:If x and y are positive integers and x + y = 3^x, is y divisible by 6?

(1) x is odd.

(2) x is a multiple of 3.

We are given that x and y are positive integers and x + y = 3^x. We need to determine whether y/6 = integer.

Recall the rules for adding even and odd numbers: (1) Odd + Odd = Even; (2) Even + Odd = Odd; (3) Even + Even = Even. Observe that the question stem can be rewritten as y = 3^x - x. Note also that 3^x is always odd. Furthermore, if x is odd, then both 3^x and x must be odd (from Rule (1)), and hence y is even. If x is even, we know that 3^x is odd, but since x is even then y must be odd.

Statement One Alone:

x is odd.

The information in statement one is not sufficient to answer the question. For instance, if x = 1, then y = 2 and y/6 IS NOT an integer. However, if x = 3, then y = 24 and y/6 IS an integer.

Statement Two Alone:

x is a multiple of 3.

The information in statement two is not sufficient to answer the question. For instance, if x = 3, then y = 24 and y/6 IS an integer. However, if x = 6, then y is an odd number and not divisible by 6.

Statements One and Two Together:

Since x is odd (from statement one), y = 3^x - x is even. Furthermore, since x is a multiple of 3 (from statement two, y = 3^x - x is a multiple of 3, since both 3^x and x are multiples of 3. Thus y is both even and a multiple 3, and such a number will be always divisible by 2 and 3, i.e., by 6. Thus y/6 is an integer.

Answer: C