Remainder problem - Does anyone know the easiest way to solve the following problem.. and also please let me know how you got the answer

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

Answer: E

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I find these types of questions (remainder) difficult, so I'll give it a shot.

Remember that...
p^2-n^2 = (p+n)(p-n)

1) Since p + n are positive integers, this tells us p+n could be: 6, 11, 16, 21, 26, 31.... (Insufficient)

2) Since we know p>n, this tells us p-n could be: 7, 4, 1, -1, -4, -7.... (Insufficient)

Taken together, we might get somewhere because (p+n)(p-n) = p^2-n^2; however, if you try a couple combinations, you'll see that a bunch of different remainders result....

(1)(11) = 11/15 = 0 +R 11
(-1)(16) = -16/16 = -1 +R -1
(-4)(11) = -44/16 = -2 +R -12

Using both of the statements together doesn't narrow down the remainder of (p^2-n^2)/15 to a single number, so the answer is E.

mdh

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

If remainder is 1 when p+n is divided by 5 then p+n = 1 or multiple of 6 (5+1)

If remainder is 1 when p+n is divided by 3 then p+n = 1 or multiple of 4 (3+1)

Combining these together you will get the answer

OOPS
I mean to say p+n is 5K+1, where K=0,1,2,3
I mean to say p+n is 3K+1, where K=0,1,2,3

wo wo, no, there are 2 significant errors here.

first of all, you know that p is greater than n. this means that (p - n) MUST be a positive number, so all the negative values you've listed are impossible.
the proper interpretation is that (p - n) does, indeed, have to be one of 1, 4, 7, ..., but also must be LESS than the listed value for p + n.
specifically, the difference between (p + n) and (p - n) is 2n, which is an even number (because n is an integer). so the two values you choose must be either both even or both odd; for instance, you can't choose p + n = 11 and p - n = 4.

still, you can prove that the answer is (e) by finding the right pair of numbers:
p + n = 11, p - n = 7 --> p^2 - n^2 = 77, remainder = 2
p + n = 6, p - n = 4 --> p^2 - n^2 = 24, remainder = 9
these two examples alone, which satisfy both (1) and (2) but yield different results, constitute conclusive proof that the answer to this problem is (e).

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second of all (this is not important for the gmat, but i figured you should have an 'fyi'):
if you're going to find remainders with negative numbers, the remainders are still positive.
this is an absolutely necessary rule; patterns don't just flip on their heads.
here's how it works for division by 3:
5 / 3: quotient = 1, remainder = 2
4 / 3: quotient = 1, remainder = 1
3 / 3: quotient = 1, remainder = 0
2 / 3: quotient = 0, remainder = 2
1 / 3: quotient = 0, remainder = 1
0 / 3: quotient = 0, remainder = 0
-1 / 3: quotient = -1, remainder = 2
-2 / 3: quotient = -1, remainder = 1
-3 / 3: quotient = -1, remainder = 0
-4 / 3: quotient = -2, remainder = 2
-5 / 3: quotient = -2, remainder = 1
-6 / 3: quotient = -2, remainder = 0
etc.

this is pretty weird, but it's the only way to define remainders that makes any sense. (it's a continuation of something called the euclidean algorithm, which is the method formally used in number theory to define remainders.)

hi ron. can you please explain the following to your answer:

the proper interpretation is that (p - n) does, indeed, have to be one of 1, 4, 7, ..., but also must be LESS than the listed value for p + n.
specifically, the difference between (p + n) and (p - n) is 2n, which is an even number (because n is an integer). so the two values you choose must be either both even or both odd;

If p and n are positive integers such that p > n, then it must be true that their difference (p - n) is less than their sum (p + n).

(p + n) - (p - n) = p + n - p + n = 2n. Therefore, the difference between p+n and p-n must be an even number. So if p+n is even, then p-n is even (because they are separated by an even number). By similar reasoning, if p+n is odd, then p-n is odd.