Hi sarthakaggarwal,
Many DS questions can be solved by TESTing VALUES - and this prompt gives us a great opportunity to do so. As with any question, you have to pay attention to the 'restrictions' that the prompt places on us, including the fact that the result of an Absolute Value calculation will NEVER be a negative number (that result will either be a positive number or a 0).
If X and Y are POSITIVE INTEGERS, is X a PRIME number?
1) |X−2| < 2−Y
2) X+Y−3 = |1−Y|
Here, we're restricted to positive integers ONLY and we're asked if X is a PRIME number. This is a YES/NO question.
1) |X−2| < 2−Y
Notice that the 'left side' of this inequality includes an Absolute Value, so that part of the calculation cannot be any smaller than 0. This is interesting, since the 'right side' must be GREATER than the 'left side', but we are SUBTRACTING a positive number from 2. This severely limits the value of Y.
Y CANNOT be 3.... since 2-3 = -1... and that won't be greater than the 'left side'
Y CANNOT be 2.... since 2-2 = 0... and that won't be greater than the 'left side'
Remember that Y MUST be a positive integer, so there's only one option left. Y MUST be 1. We can 'lock' that value in place.
Y = 1
Now we have...
|X-2| < 1
Again, we're restricted to positive integers only for X - and there's only one possible value for X that will fit this inequality: it's when X = 2. This would give us...
|2-2| < 1
0 < 1
Since X = 2, the answer to the question is YES... and since there's only one possible answer, the answer is really ALWAYS YES.
Fact 1 is SUFFICIENT.
Notice how no special calculations were needed here; just a bit of Arithmetic and TESTing VALUES to determine what options are possible. Fact 2 will require a little more work, but the type of work will be the same.
2) X+Y−3 = |1−Y|
Here, the 'right side' of the equation can never be any smaller than 0, but the 'left side' includes a "-3", so we have to be sure that the result of that part of the calculation never drops below 0. Let's try TESTing some values and see what happens...Remember that both X and Y must be POSITIVE INTEGERS.
IF.... Y = 1
X+1-3 = |1-1|
X - 2 = 0
X = 2... Here, the answer to the question is YES.
IF.... Y = 2
X+2-3 = |2-1|
X - 1 = 1
X = 2... Here, the answer to the question is YES.
IF.... Y = 3
X+3-3 = |3-1|
X = 2
X = 2... Here, the answer to the question is YES.
Notice the pattern? If you're not sure yet, then feel free to try a few more TESTs (since you're just doing basic Arithmetic, that additional work shouldn't take too long. The answer to the question is ALWAYS YES.
Fact 2 is SUFFICIENT
Final Answer:
GMAT assassins aren't born, they're made,
Rich