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.
If x and y are positive integers and x + y = 3^x, is y divis
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We have to see whether y divisible by 6.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 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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y will be a multiple of 6 if it is both EVEN and DIVISIBLE BY 3.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.
3^x = odd multiple of 3.
Rephrasing x + y = 3^x, we get:
y = 3^x - x = (odd multiple of 3) - x.
Since ODD - ODD = EVEN, y will be even if the value in blue is ODD.
Since (multiple of 3) - (multiple of 3) = multiple of 3, y will be a multiple of 3 if the value in blue is a multiple of 3.
Question stem, rephrased:
Is x an ODD MULTIPLE OF 3?
Statement 1:
Not way to determine whether x is a multiple of 3.
INSUFFICIENT.
Statement 2:
No way to determine whether x is odd.
INSUFFICIENT.
Statements combined:
Thus, x is an odd multiple of 3.
SUFFICIENT.
The correct answer is C.
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We are given that x and y are positive integers and x + y = 3^x. We need to determine whether y/6 = integer.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.
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
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