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Math Expert V
Joined: 02 Sep 2009
Posts: 60555
If x and y are positive integers, is x divisible by 3?  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 61% (02:04) correct 39% (01:47) wrong based on 48 sessions

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If x and y are positive integers, is x divisible by 3?

(1) x - y is divisible by 3

(2) x - 2y is divisible by 3

Are You Up For the Challenge: 700 Level Questions

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Senior Manager  P
Joined: 25 Jul 2018
Posts: 471
If x and y are positive integers, is x divisible by 3?  [#permalink]

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If x and y are positive integers, is x divisible by 3?

(Statement1): x - y is divisible by 3
6–3 =3 —> (yes)
5 —2= 3 —> (No)
Insufficient

(Statement2): x - 2y is divisible by 3
18 —2*3= 12 (yes)
5 —2*1= 3 (No)
Insufficient

Taken together 1&2,

If both numbers are divisible by 3, then adding and subtracting those numbers from each other are also divisible by 3.

x — y —(x—2y) = x—y—x+ 2y= y
—> y is divisible by 3.

In order x—y to be divisible by 3, x must be divisible by 3.
Sufficient

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Re: If x and y are positive integers, is x divisible by 3?  [#permalink]

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Bunuel wrote:
If x and y are positive integers, is x divisible by 3?

(1) x - y is divisible by 3

(2) x - 2y is divisible by 3

Are You Up For the Challenge: 700 Level Questions

#1
x - y is divisible by 3
x=5 , y=2 ; YES
x=8, ,y=2 ; NO
#2
x - 2y is divisible by 3
x=8 , y = 1 ; NO
x=9 , y=3 ; yes
from 1 &2
x=8 is common ; NO is condition
IMO C
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8436
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: If x and y are positive integers, is x divisible by 3?  [#permalink]

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Bunuel wrote:
If x and y are positive integers, is x divisible by 3?

(1) x - y is divisible by 3

(2) x - 2y is divisible by 3

Are You Up For the Challenge: 700 Level Questions

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have 2 variables (x and y) and 0 equations, C is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
We have x - y = 3m and x - 2y = 3n for integers m and n.
Then 2x - 2y = 6m and x=(2x-2y)-(x-2y) = 6m-3n = 3(2m-n).
It means x is divisible by 3 and the answer is 'no'.

Since both conditions together yield a unique solution, they are sufficient.

Since this question is an integer question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
If x = 6 and y = 3, then x - y is divisible by 3 and x is divisible by 3, which means the answer is 'yes'
If x = 5 and y = 2, then x - y is divisible by 3 and x is not divisible by 3, which means the answer is 'no'
Since condition 1) does not yield a unique solution, it is not sufficient

Condition 2)
If x = 6 and y = 3, then x - 2y is divisible by 3 and x is divisible by 3, which means the answer is 'yes'
If x = 5 and y = 4, then x - 2y is divisible by 3 and x is not divisible by 3, which means the answer is 'no'
Since condition 2) does not yield a unique solution, it is not sufficient

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
_________________ Re: If x and y are positive integers, is x divisible by 3?   [#permalink] 07 Nov 2019, 11:03
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