BANON wrote:
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.
\(p^n-n^2 = (p+n)(p-n)\)
Question stem, rephrased:
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 1In other words, p+n is 1 more than a multiple of 5:
p+n = 5a+1 =
6, 11, 16, 21, 26...Statement 2: The remainder when p - n is divided by 3 is 1In other words, p-n is 1 more than a multiple of 3:
p-n = 3a+1 =
1, 4, 7, 10, 13, 16...It seems unlikely that either statement on its own will be sufficient to determine the remainder when (p+n)(p-n) is divided by 15.
Test whether different remainders are possible when the statements are combined.
(p+n) + (p-n) = 2p = EVEN.
Implication of the equation above:
When the statements are combined, p will be a positive integer such that p>n if a value from the blue list in Statement 1 is combined with a smaller value from the red list in Statement 2 such that their sum is EVEN.
Case 1:
p+n = 6 and
p-n = 4, with the result that (p+n)(p-n) = 6*4 = 24
In this case, dividing (p+n)(p-n) by 15 yields a remainder of 9.
Case 2:
p+n = 11 and
p-n = 1, with the result that (p+n)(p-n) = 11*1 = 11
In this case, dividing (p+n)(p-n) by 15 yields a remainder of 11.
Since different remainders are possible, the two statements combined are INSUFFICIENT.
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