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

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

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New post 07 Nov 2019, 02:45
00:00
A
B
C
D
E

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?  [#permalink]

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New post 07 Nov 2019, 08:07
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

The answer is C

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

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New post 07 Nov 2019, 10:22
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


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

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New post 07 Nov 2019, 11:03
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

Therefore, C is the answer.

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.
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Re: If x and y are positive integers, is x divisible by 3?   [#permalink] 07 Nov 2019, 11:03
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