If p and n are positive integers and p > n, what is the remainder when p² - n² 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.

Target question: What is the remainder when p² - n² is divided by 15

NOTE that p² - n² is a difference of squares, so we can FACTOR it to get: p² - n² = (p + n)(p - n). Since both (p + n) and (p - n) are in the statements, it may be useful to REPHRASE the target question...

REPHRASED target question: What is the remainder when (p + n)(p - n) is divided by 15?

Statement 1: The remainder when (p + n) is divided by 5 is 1
This tell us that (p + n) is NOT DIVISIBLE by 5.
Since there's no information about (p-n), we can't determine the remainder when (p + n)(p - n) is divided by 15
Consider these two conflicting cases:
Case a: p = 5 and n = 1 (notice that the remainder when p+n is divided by 5 is 1). In this case, the remainder when is 9 when (p + n)(p - n) is divided by 15
Case b: p = 1 and n = 0 (notice that the remainder when p+n is divided by 5 is 1). In this case, the remainder when is 1 when (p + n)(p - n) is divided by 15
So, 2 of the numbers are less than 30
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The remainder when p - n is divided by 3 is 1
Here we have no information about p+n.
Consider these two conflicting cases:
Case a: p = 5 and n = 1 (notice that the remainder when p-n is divided by 3 is 1). In this case, the remainder when is 9 when (p + n)(p - n) is divided by 15
Case b: p = 1 and n = 0 (notice that the remainder when p-n is divided by 3 is 1). In this case, the remainder when is 1 when (p + n)(p - n) is divided by 15
So, 2 of the numbers are less than 30
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
IMPORTANT: Notice that I happened to use the same values for the counter-examples in each statement. This means that we can use the same values here to show that the COMBINED statements are not sufficient. That is...
Consider these two conflicting cases:
Case a: p = 5 and n = 1 (notice that both statements are satisfied). In this case, the remainder when is 9 when (p + n)(p - n) is divided by 15
Case b: p = 1 and n = 0 (notice that both statements are satisfied). In this case, the remainder when is 1 when (p + n)(p - n) is divided by 15
So, 2 of the numbers are less than 30
Since we cannot answer the target question with certainty, the COMBINED statements are NOT SUFFICIENT

Answer: E

----------------------------

ALTERNATIVELY, when examining the statements combined, we can use a nice rule that says:
If N divided by D, leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

Okay, onto the question . . .
Statement 1: Applying the above rule, some possible values of p+n are 6, 11, 16, 21, 26, etc.
Aside: you'll notice that I didn't include 1 as a possible value since we're told that p and n are positive integers, and we can't get a sum of 1 if both are positive

Statement 2: Applying the above rule, some possible values of p-n are 1, 4, 7, 10, 13, etc

Let's examine two cases with conflicting results.

case a: p+n = 11 and p-n = 1
Add the equations to get 2p = 12, which means p = 6 and n = 5 (perfect, we have positive integer values for p and n)
In this case, when (p + n)(p - n) is divided by 15, the remainder is 11

case b: p+n = 6 and p-n = 4
Add the equations to get 2p = 10, which means p = 5 and n = 1 (perfect, we have positive integer values for p and n)
In this case, when (p + n)(p - n) is divided by 15, the remainder is 9
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer = E

Cheers,
Brent