What is the remainder when the positive integer n is divided by 5 ?

This question primarily hinges on our understanding of the relationship between dividend, divisor, quotient, and remainder. If we can either solve for n, or solve for the units digit of n, we will have sufficiency (as divisibility by 5 depends entirely on the units digit). Now we're ready to take a look at our statements!

Statement (1) tells us that "When n is divided by 3, the quotient is 4 and the remainder is 1." If we set this up mathematically using our understanding that the dividend = (divisor)*(quotient) + remainder, we can solve for "n," as n = 3*4 + 1, or 13. If we know "n," we certainly know the remainder when we divide "n" by 5 - in this case, 3. (Sufficient)

With Statement (2), we know that n/4 gives us a remainder of 1, so we know that n = "some multiple of 4, plus 1." However, we can quickly disprove sufficiency, as we could have 4+1 = 5, giving us a remainder of 0, or 8+1 = 9, giving us a remainder of 4. As soon as we can pick permissible values that give us different values for this value Data Sufficiency quesiton, we have disproven sufficiency. (Not Sufficient)

In this case, taking a moment to preempt what we need to have sufficiency, and recognizing that if we're given enough information to solve for "n," we have sufficiency, and that if we are able to quickly choose permissible values for "n" that give us different answers to our value DS question, we can efficiently disprove sufficiency, we are left with (A).