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Divisible by 11?

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Source: — Data Sufficiency |

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by [email protected] » Fri Sep 16, 2016 6:22 pm
Hi carlos.lara.7,

This question can be solved by TESTing VALUES, but it involves a relatively rare Number Property that most Test Takers would not recognize. We're told that X and Y are NON-ZERO INTEGERS. We're asked if X is divisible by 11. This is a YES/NO question.

1) 2X + 8 = 3Y

IF...
X = 2, Y = 4
then the answer to the question is NO

IF...
X = 11, Y = 10
then the answer to the question is YES
Fact 1 is INSUFFICIENT

2) 6Y - 5 is divisible by 11.

This tells us NOTHING about X.
Fact 2 is INSUFFICIENT

Combined, we know.
2X + 8 = 3Y
6Y - 5 is divisible by 11.

Notice how Fact 1 uses '3Y' and Fact 2 uses '6Y' - we can combine those facts. First we 'double' the first equation...

4X + 16 = 6Y
6Y - 5 = positive multiple of 11

4X + 16 - 5 = positive multiple of 11
4X + 11 = positive multiple of 11

Since we're adding two terms together, and one of the terms (the '11') is a multiple of 11, for the SUM to be a multiple of 11 the OTHER term (the "4X") must ALSO be a multiple of 11. Since X is an integer, the only way for 4X to a multiple of 11 is if X itself is a multiple of 11. Under these conditions, the answer to the question is ALWAYS YES.
Combined, SUFFICIENT.

Final Answer: C

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by Jay@ManhattanReview » Tue Dec 20, 2016 5:27 am
carlos.lara.7 wrote:If x and y are non-zero integers, is x divisible by 11?

(1) 2x+8 = 3y

(2) 6y-5 is divisible by 11
S1: We have no clue about the value of y, so we cannot deduce whether x is divisible by 11.

We can merely manipulate 2x+8 = 3y to x = (3y - 8)/2. Insufficient.

S2: We have no clue about the value of x, so we cannot deduce whether x is divisible by 11.

S1 and S2: Let us manipulate S1 equation such that we are able to exploit S2 information.

S1 equation: x = (3y - 8)/2 = 2*(3y - 8)/(2*2) = (6y - 16)/4 = (6y - 5 - 11)/4

=> x = (6y - 5)/4 - 11/4

Since we know from S2 that (6y - 5) is divisible by 11, thus the first term (6y - 5)/4 is divisible by 11; needless to prove that 11/4 is divisible by 11. :) Sufficient.

OA: C

Hope this helps!

-Jay

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