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

p^2 - n^2 simplifies to (P+n)(P-n) dont both 1 and 2 together answer the question?

I think the answer is E
Take numbers: p=6,n=5 or P=10,n=6, the reminders are different even they satisfy 1) and 2)

what is the offical answer for this?

we separate p-n and p+n and divide them individually by the denominators 3 and 5 respectively, however we surely cannot multiply the remainders to get to get a consistent answer. please plug in to confirm this. Answer should be E. Pls. confirm.

both statements taken together answer the question

(P+N)/5=Integer(A) + 1/5-statement 1
(P-N)/3=Interger(B) + 1/3-statemetn 2
multiply 1 and 2
(P^2-N^2)/15=(5A+1)/5 * (3B+1)/3=(15AB(Integer)+5A(integer)+3B(integer)+1)/15
P^2-N^2=15AB(Integer)+5A(integer)+3B(integer)+1=Interger
and so the remainder is 0

Instructors ... how would you recommend we approach this DS problem. An alternate method to the one above would help.

the first thing you should notice here is that the polynomial in the question stem (p^2 - n^2) is equivalent to (p + n)(p - n). nicely enough, these happen to be the exact expressions in the two choices.
therefore:
the key to this problem is to consider (p + n) and (p - n) as SINGLE QUANTITIES, not as a sum and a difference.

the other key is to realize that remainders are ALWAYS ALWAYS ALWAYS based on repetition, and so:
you can solve just about any data sufficiency problem about REMAINDERS by simply MAKING LISTS of the numbers that satisfy the statements, and then plugging those numbers into the question stem.

in fact, lists are SO reliable for solving remainder problems that, if i (ron) personally see a data sufficiency problem involving remainders, i will go STRAIGHT to making lists, no ifs, ands, or buts, no questions asked.

viz.:

statement (1)
(p + n) could be
1
6
11
16
21
26
etc.

statement (2)
(p - n) could be
1
4
7
10
13
16
19
etc.

if you have both of these together, just try a bunch of random combinations. if you keep coincidentally getting the same answer every time, then eventually you'll surrender and pick "sufficient", but, if at any time you get two divergent answers, then you're done -- you've just proved "insufficient".

combining these two statements:
(p - n)(p + n) could be
6 x 4 = 24 --> remainder is 9
6 x 7 = 42 --> remainder is 12
whoa!
done.
insufficient.
ans (e)