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.

okay here is my question
P^2 - n^2 = (p+n)(p-N)
question ask that what is the remainder when p^2 - n^2 is divisible by 15
we know from statement 1) that when (p+n)/5 the remainder is 1 and when (p-n)/3 the remainder is 1. we know that 5.3= 15. so isnt the two statement enough to say that the remainder is 2.

sana.noor wrote:okay here is my question
P^2 - n^2 = (p+n)(p-N)
question ask that what is the remainder when p^2 - n^2 is divisible by 15
we know from statement 1) that when (p+n)/5 the remainder is 1 and when (p-n)/3 the remainder is 1. we know that 5.3= 15. so isnt the two statement enough to say that the remainder is 2.

Sana, the statement 2 says (p-n)/5 and not (p-n)/3.

Also, The question doesn't say that p+n and p-n are completely divisible; the rule you are trying to forumulate is wrong.

The logic you are trying to workout doesnt work for multiplication.

For example:
Assume question stated (p-n)/3

Then Let p+n = 26 ->r=1 & p-n = 4 ->r=1 --> 26x4/15 = remainder = 14 NOT EQUAL TO 2