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.
I'm happy to help with it here.
The first skill we need is the "
rebuilding the dividend" equation. See:
https://magoosh.com/gmat/2012/gmat-quant ... emainders/If we divide N by D, and get a quotient of Q with remainder of R, then
N/D = Q + (R/D)
or
N = Q*D + RThat is one of the most powerful and unappreciated formulas on the GMAT, the "rebuilding the dividend" equation. As for the name, the terminology here is
N = the dividend, the thing divided
D = the divisor, the number by which we are dividing
Q = the quotient, the result of division
R = the remainder
Here, we can change the statements into equations:
Statement #1: (p + n) = 5*S + 1, for some quotient integer S
Statement #2: (p - n) = 3*T + 1, for some quotient integer T
The second thing that you really need to recognize quickly on the GMAT is the
Difference of Two Squares patterns, the single most important algebraic pattern on the entire GMAT. See:
https://magoosh.com/gmat/2012/gmat-quant ... o-squares/https://magoosh.com/gmat/2013/three-alge ... -the-gmat/Here, this implies that:
p^2 - n^2 = (p + n)*(p - n)
One statement talks only about 5, and the other, only about 3, so neither is sufficient individually. If both are considered together, we can multiply the two formulas together:
p^2 - n^2 = (p + n)*(p - n) = (5*S + 1)*(3*T + 1) = 15*ST + 5*S + 3*T + 1
Well, clearly the first term, (15*ST) would be divisible by 15. The problem is: we don't know the values of S or T. We know they are integers, but we don't know their values. If think about possible values of 5*S, multiples of 5, then when we divide multiples of 5 by 15, we could have remainders of 0, 5, or 10. Similarly, when we divided 3*T, multiples of 3, by 15, we could have remainders of 0, 3, 6, 9, or 12. Thus, 15 will go evenly into the first term, but may have a remainder on the second term and on the third time, and of course, will have a remainder of 1 on the last term. Because there's still choice about the possible remainders, even with both statements combined, we cannot answer the prompt question.
Even with both statements combined, everything is still insufficient. Answer =
(E)Does all this make sense?
Mike