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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