If p and n are positive integers and p>n, what is the remainder when p^2 - n^2 is divided by 15?
First of all p^2 - n^2=(p+n)(p-n)
(1) The remainder when p + n is divided by 5 is 1. No info about p-n. Not sufficient.
(2) The remainder when p - n is divided by 3 is 1. No info about p+n. Not sufficient.
(1)+(2) "The remainder when p + n is divided by 5 is 1" can be expressed as p+n=5t+1
and "The remainder when p - n is divided by 3 is 1" can be expressed as p-n=3k+1
Multiply these two --> (p+n)(p-n)=(5t+1)(3k+1)=15kt+5t+3k+1
, now first term (15kt) is clearly divisible by 15 (r=0), but we don't know about 5t+3k+1. For example t=1 and k=1, answer r=9 BUT t=7 and k=3, answer r=0. Not sufficient.
OR by number plugging: if p+n=11
(11 divided by 5 yields remainder of 1) and p-n=1
(1 divided by 3 yields remainder of 1) then (p+n)(p-n)=11
and remainder upon division 11 by 15 is 11 BUT if p+n=21
(21 divided by 5 yields remainder of 1) and p-n=1
(1 divided by 3 yields remainder of 1) then (p+n)(p-n)=21
and remainder upon division 21 by 15 is 6. Not sufficient.
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