p^2-n^2 can be written as (p-n)(p+n) when divided by 15
--> {(p-n)/3}{(p+n)/5}
A and B alone are insufficient as they do not provide information about the other factor
try combining both of these
(p-n)=3s+1 [Assume s as any integer]
(p+n)=5t+1 [Assume t as any integer]

(3s+1)(5t+1)/15 =>(15st+3s+5t+1)/15
so until we have info about s and t we can't find the remainder


Hence E

fangtray wrote:If p and n are positive integers and p > n, what is the remainder when p^2-n^2 is divided by 15?

1. The remainder when p+n is divided by 5 is 1
2. The remainder when p-n is divided by 3 is 1

Question rephrased: What is the remainder when (p+n)(p-n) is divided by 15?

Statement 1: No information about p+n. INSUFFICIENT.
Statement 2: No information about p-n. INSUFFICIENT.

Statements 1 and 2 combined:
The remainder when p+n is divided by 5 is 1.
This statement implies the following:
p+n = 5k+1, where k≥0.
Thus, p+n = 1,6,11,16,21...

The remainder when p-n is divided by 3 is 1.
This statement implies the following:
p-n = 3m+1, where m≥0.
Thus, p-n = 1,4,7,10,13...

Case 1: p+n=11 and p-n=1
Adding the two equations:
2p=12
p=6, implying than n=5.
(p+n)(p-n)/15 = (11*1)/15 = 0 R11.

Case 2: p+n=21 and p-n=1
Adding the two equations:
2p=22
p=11, implying than n=10.
(p+n)(p-n)/15 = (21*1)/15 = 1 R6.

Since different remainders are possible, INSUFFICIENT.

The correct answer is E.

IMO: E