**Quote:**

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

a) The reminder when p + n is divided by 5 is 1.

b) The reminder when p - n is divided by 3 is 1.

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a): p+n/5 has a remainder of 1. Therefore, p+n = 6 or p+n+11 or p+n = 16 (so on and so forth)

Let's assume p + n = 6

Substitute 5 for p and 1 for n (because P>N)

Also, substitute 4 for p and 2 for n

Take p=5 and n = 1

5^2 - 1^2 = 24 and 24/15 has a remainder of 9

Take p = 4 and n = 2

4^2 - 2^2 = 16-4 = 12 and 12/15 has a remainder of 12

Therefore A is insufficient

b) p-n/3 has a remainder of 1 so p-n = 4 or p-n =7 or p-n = 11

If p-n = 4, p-n = 5-1 or 6-2 and so on

If p = 5 and n = 1 then

5^2 - 1^2 = 24 and 24/15 has a remainder of 9

If p = 6 and n = 2 then

6^2 - 2^2 = 32 and 32/15 has a remainder of 2

Therefor B is insufficient

Take both together:

Well you can assume P = 5 and n = 1 (because 5-1 = 4 and 5+1 = 6; using the information from above)

You already worked the math and figured that the remainder would be 9

However, you can have P = 9 and N = 2

9+2 = 11 (which satisfies part A) and 9 -2 = 7 (which satisfies part B)

9^2 - 2^2 = 81 - 4 =77

77/15 has a remainder of 2

Therefore, I'm getting the answer is E