If p and n are positive integers p >n , what is the remainder when p^2-n^2 is divided by 15
The remainder when p+q is divided by 5 is 1
The remainder when p-q is divided by 3 is 1





essential fact: this problem features the DIFFERENCE OF SQUARES, p^2 - n^2 = (p - n)(p + n). if you don't make this realization IMMEDIATELY, then you're going to be mired forever in trying to solve for p and n individually. (remember that, if a problem is stated entirely in terms of combinations, then the problem can almost certainly be solved entirely in terms of combinations.)
lionking's first example chooses p + n = 11 (which gives a remainder of 1 upon division by 5, as required), and p - n = 1 (which gives a remainder of 1 upon division by 3, as required).
when 11 x 1 = 11 is divided by 15, the remainder is 11.
lionking's second example chooses p + n = 6 (which gives a remainder of 1 upon division by 5, as required), and p - n = 1 (which gives a remainder of 1 upon division by 3, as required).
when 6 x 1 = 11 is divided by 15, the remainder is 6.
these are two different remainders, so, insufficient.
Ans - E