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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.
The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.
3^1 ~ 3^5 ~ 3^9 ~ ... ~ 3 : Integers of the form 3^{4x+1} always have the remainder of 3 when they are divided by 10.
3^2 ~ 3^6 ~ 3^10 ~ ... ~ 9 : Integers of the form 3^{4x+2} always have the remainder of 9 when they are divided by 10.
3^3 ~ 3^7 ~ 3^11 ~ ... ~ 7 : Integers of the form 3^{4x+3} always have the remainder of 7 when they are divided by 10.
3^4 ~ 3^8 ~ 3^12 ~ ... ~ 1 : Integers of the form 3^{4x} always have the remainder of 1 when they are divided by 10.
Therefore, the remainder when 3^{4x + 1} + y is divided by 10 depends only on the value of y.

Only condition 2) gives us a value for y.
Therefore, B is the answer.
Answer: B

Max@Math Revolution wrote:
If x and y are positive integers, what is the remainder when 3^{4x+1}+y is divided by 10?

1) x=2
2) y=3

Recall that the remainder when a number is divided by 10 depends on the units digit of the number. Thus if we can determine the units digits of 3^{4x+1} + y, we can determine the remainder (since it will just be the units digit).

Statement One Alone:

x = 2

Therefore, 3^{4x+1} + y = 3^9 + y. Without knowing the value of y, we can't determine the units digit of 3^{4x+1} + y. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

y = 3

Therefore, 3^{4x+1} + y = 3^{4x + 1} + 3. It seems the statement is not sufficient since we don't know the value of x. However, we may recall that the base of 3, has a repeating units digit pattern of 3-9-7-1 when it is raised to a positive integer power. Thus, 3 raised to a power that is a multiple of 4 will always end in a 1 and furthermore 3 raised to a power that is 1 more than a multiple of 4, will always end in a 3.

Thus, without even knowing the value of x, we can determine that 3^(4x+1) will always end in a 3. Therefore, the units digit of 3^{4x+1} + y = 3 + 3 = 6. Statement two alone is sufficient to answer the question.

Answer: B

Statement 1= Does not provide information about the value of y it only gives us information about x only, we need the value of y to determine the unit digits of the equation given, so statement 1 is insufficient

Statement 2= $$y=3^{\left(4x+1\right)}$$ we have its unit digit as 3 and it will definitely because 6, when 3 is added to it, so statement 2 is sufficient .
answer = option B because statement 2 alone is SUFFICIENT but statement 1 alone is NOT