Fastest way

Yes, there is.

If the remainder of a number divided by 10 has to be 0, then the number has to be zero..

which means the value of 3^x + 1 must end with 0..or which means the value of 3^x must end with 9.

So, we should know the cyclicity of number 3 which is 4.

If you note the series, u can see that it is 3,9,27,81,243,729....

So, 2nd, 6th, 10th, 14th digit would all end with a value of 9.

All these informations are pre-requisites to attend this problem which would help us to solve this one within 30 seconds.

Now, look at at the two statements..

statement 1 says X = 4n + 2, where n is a positive integer...

substitute any value for n and u will find that X will end up with values like 6, 10, 14 etc...

which means 3^x will end with 9 and 3^x + 1 with 10..remainder 0...

Sufficient..

Statement 2 says X > 4...X can be any number...So statement 2 alone is insufficient..

Hence, the answer is A.

Here's another approach.
We have to chk if 3^x + 1 is divisible by 10.

From 1 we have x=4n+2 or x=2(2n+1) which means x is a even multiple of an odd number(2n+1 is an odd number). as we are given n is a positive integer so min value of n is 1. so min value of x we get is 6 which is indeed suff.

From 2 we get x>4
when x=5
3^5 = 243
243+1 = 244 is not divisible by 10 and 3^6 + 1 is divisible by 10.
hence b is insuff.
Hence ans is A.