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

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.
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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.

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

My attempt:
(1) x - y is divisible by 3
x-y = 3*a (where a is a stand-in for multiple of 3)
(X) INSUFFICIENT

(2) x - 2y is divisible by 3
x-2y = 3*b (where b is a stand-in for a multiple of 3)
Also (X) INSUFFICIENT

Combined:
x-y = 3*a
x-2y = 3*b

Multiply the first equation by 2 = 2x-2y=6*a and stack both equations to minus. This lets you remove y from the equations
2x-2y=6*a
- x-2y=3*b
---------------
x=6*a-3*b
x=3(2*a-b)
By noticing a and b are both multiples of 3 you can confirm x is divisible by 3

Answer: C

If you understand the concept of divisibility and grouping, this can be done in a few seconds.

(1) x-y is divisible by 3.

Just because the difference is divisible doesn't mean the numbers are multiples of 3 too. e.g. 8-5 = 3.

(2) x-2y is divisible by 3.

Same as above. e.g. x = 8 and y = 1 so x - 2y = 8 - 2 = 6

Using both, when we subtract y out of x, we are left with a multiple of 3. When we subtract another y, we are again left with a multiple of 3. This means y must be a multiple of 3 too since you must have taken out groups of 3 when you took out y. Then x MUST be a multiple of 3 too since x is made by adding a multiple of 3 to y which is a multiple of 3 too.

Answer (C)

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