if p and n are positive ints 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 reminder when p - n is divided by 3 is 1

IMO E.



Thanks!

Can someone please tell the steps for the same.

if p and n are positive ints 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 reminder when p - n is divided by 3 is 1

Rephrase question as : (p+n) (p-n) /15 leaves what remainder?

1) (p+n) divided by 5 leaves 1. So (p+n) can be 6,11,16 so on

Insufficient

2) (p-n) divided by 3 leaves 1. So (p-n) can be 4,7,10 so on

Insufficient

Now (p+n) (p-n) leaves remainder what when divided by 15. Note it has to work for ALL numbers.

6*4/15 leaves 9

6*7/15 leaves 12

You got two answers. So clearly insufficient with C also.

Hence E

PGSD - well played.

two things:

--

(1)
if you have a problem about REMAINDERS, you should view that problem as an opportunity for PATTERN RECOGNITION.
there are lots of topics that lead to recognition of common patterns - i.e., remainders aren't the only topic of such problems - but, in remainder problems, CLEAR patterns tend to emerge QUICKLY if you start testing numbers in some sort of systematic manner.

in this problem, therefore, and in problems like it:
if you don't immediately see a better technique, you should JUST START PLUGGING IN SAMPLE NUMBERS AND LOOK FOR A PATTERN.

do not kill yourself trying to apply theory to a stubborn problem that won't yield. there are no points for style on this test.
instead, if you can't open the problem with theory pretty much immediately, then try something else.

--

(2)
if you didn't recognize AND factor the difference of squares in this problem, p^2 - n^2 --> (p+n) (p-n), RIGHT AWAY, then you MUST put that sort of thing on a flash card, so that you can INSTANTLY recognize it next time. that is the single most important factoring pattern in all of algebra, so you must know it.