The question is asking "x is how much larger than a multiple of 4". If, from Statement 1, we know x+1 is 3 greater than a multiple of 4, then x itself must be 2 greater than a multiple of 4, so Statement 1 is sufficient.

Statement 2 tells us that 2x is divisible by 4, but that will be true any time x is even. So the remainder can be 0 or 2 when we divide x by 4, and Statement 2 is not sufficient.
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Target question: What is the remainder when x is divided by 4 ?

Statement 1: The remainder is 3 when x + 1 is divided by 4.
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There's a nice rule that says, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3
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We don't know how many times 4 divides into x+1. So, let's just say 4 divides into x+1 k times.
So, we can write: x + 1 = 4k + 3 (for some integer k)
Subtract 1 from both sides to get x = 4k + 2
Since 4k is a multiple of 4, we can see that x is 2 greater than some multiple of 4
So, when we divide x by 4, the remainder will be 2
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The remainder is 0 when 2x is divided by 4.
We can write: 2x = 4k (for some integer k)
Divide both sides by 2 to get: x = 2k
This tells us that x is an even integer.
There are several values of x that satisfy statement 2. Here are two:
Case a: x = 2. In this case, the answer to the target question is when we divide x by 4, the remainder will be 2
Case b: : x = 4. In this case, the answer to the target question is when we divide x by 4, the remainder will be 0
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

RELATED VIDEO FROM MY COURSE

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Hi All,

We're told that X is an INTEGER GREATER than 0. We're asked for the remainder when X is divided by 4. This question is based around some basic Arithmetic rules (and you can TEST VALUES to define the patterns involved). To start, here are the first several positive integers - and what happens when you divide by 4...

1/4 = 0r1
2/4 = 0r2
3/4 = 0r3
4/4 = 1r0
5/4 = 1r1
6/4 = 1r2
7/4 = 1r3
8/4 = 2r0
Etc.

(1) The remainder is 3 when (X + 1) is divided by 4.

Based on the pattern defined above, if we get a remainder of 3 when dividing (X+1) by 4, then removing the "+1" would move us one position 'up' in the pattern - meaning that the remainder when X/4 will ALWAYS equal 2. You can see this in a couple of examples:
IF...
X = 2, then (2+1)/4 = 0r3 and 2/4 = 0r2
X = 6, then (6+1)/4 = 1r3 and 6/4 = 0r2
Etc.
Fact 1 is SUFFICIENT

(2) The remainder is 0 when 2x is divided by 4.

Fact 2 asks us to focus on values that would create a remainder of 0. That will occur with any multiple of 4. However, Fact 2 asks us to consider "2X" and that does not necessarily mean that X is a multiple of 4...
IF...
X = 2, then (2)(2)/4 = 1r0 and 2/4 = 0r2
X = 4, then (2)(4)/4 = 2r0 and 4/4 = 1r0
Thus, the answer to the question could be 2 OR 0.
Fact 2 is INSUFFICIENT

Final Answer:

GMAT assassins aren't born, they're made,
Rich
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The remainder is 3 when x + 1 is divided by 4.

x + 1 can be values such as 3, 7, 11, 15, and so forth. Thus, x is 2, 6, 10, 14, etc..

When any of those values of x is divided by 4, the remainder is 2.

Statement one alone is sufficient to answer the question.

Statement 2 :

The remainder is 0 when 2x is divided by 4.

2x can be values such as 4, 8, 12 ,16, and so forth. Thus, x is 2, 4, 6, 8, etc.

We see that 2/4 = 0 remainder 2 and 4/4 = 1 remainder 0.

Statement two alone is not sufficient to answer the question.

Answer: A