DS question

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DS question

by Nijo » Wed Aug 13, 2014 8:29 am
If p and n are positive integers and p>n, what is the remainder when p^2 - n^2 is divided by 15?
1) when p+n is divided by 5, the remainder is 1
2) when p-n is divided by 3 the remainder is 1

OA is E

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by [email protected] » Wed Aug 13, 2014 9:04 am
Hi Nijo,

This question is perfect for TESTing Values.

We're told that P and N and POSITIVE INTEGERS and P>N. We're asked for the REMAINDER when P^2 - N^2 is divided by 15?

*If you find it easier, you can rewrite P^2 - N^2 as (P+N)(P-N)*

Fact 1: (P+N)/5 has a remainder of 1

This Fact means that (P+N) could be 6, 11, 16, 21, etc.

Let's TEST Values....
If... P=5, N=1, then 25-1 = 24.......24/15 = 1r9 so the answer to the question is 9
If....P=4, N=2, then 16-4 = 12.......12/15 = 0r12 so the answer to the question is 12
Fact 1 is INSUFFICIENT

Fact 2: (P-N)/3 has a remainder of 1

This Fact means that (P-N) could be 4, 7, 10, etc.

Let's TEST Values...

If...P=5, N=1.....then the answer to the question is 9 (the proof is in Fact 1)
If...P=6, N=2, then 36-4 = 32......32/15 = 2r2 and the answer to the question is 2
Fact 2 is INSUFFICIENT

Combined we know that...
(P+N) could be 6, 11, 16, 21, etc.
(P-N) could be 4, 7, 10, etc.

We already know that with P=5, N=1 that the answer would be 9, so if we can find ANY other possibility, we'll be done.
If P=10, N=6, then 100-36=64.....64/15 = 4r4 and the answer would be 4
Combined, INSUFFICIENT

Final Answer: E

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by GMATGuruNY » Wed Aug 13, 2014 9:10 am
Nijo 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) when p+n is divided by 5, the remainder is 1
2) when p-n is divided by 3 the remainder is 1
Statement 1:
In other words, p+n a (MULTIPLE OF 5) + 1.
Thus:
p+n = 5a + 1 = 1, 6, 11, 16, 21...

Let p+n = 6.
If p=5 and n=1, then p²-n² = 24, in which case dividing by 15 will yield a remainder of 9.
If p=4 and n=2, then p²-n² = 12, in which case dividing by 15 will yield a remainder of 12.
Since the remainder can be different values, INSUFFICIENT.

Statement 2:
In other words, p-n is a (MULTIPLE OF 3) + 1.
Thus:
p-n = 3b + 1 = 1, 4, 7, 10, 13, 16...

Let p-n= 4.
If p=5 and n=1, then p²-n² = 24, in which case dividing by 15 will yield a remainder of 9.
If p=6 and n=2, then p²-n² = 32, in which case dividing by 15 will yield a remainder of 2.
Since the remainder can be different values, INSUFFICIENT.

Statements combined:
Statement 1: p+n = 1, 6, 11, 16, 21...
Statement 2: p-n = 1, 4, 7, 10, 13, 16...

If p+n=6 and p-n=4, then p=5 and n=1.
Here, p²-n² = 24, in which case dividing by 15 will yield a remainder of 9.
If p+n=11 and p-n=1, then p=6 and n=5.
Here, p²-n² = 11, in which case dividing by 15 will yield a remainder of 11.
Since the remainder can be different values, INSUFFICIENT.

The correct answer is E.
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by Brent@GMATPrepNow » Wed Aug 13, 2014 2:11 pm
If p and n are positive integers and p > n, what is the remainder when p² - n² 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.
Target question: What is the remainder when p² - n² is divided by 15

NOTE that p² - n² is a difference of squares, so we can factor it to get: p² - n² = (p + n)(p - n). Since both (p + n) and (p - n) are in the statements, it may be useful to REPHRASE the target question...

Rephrased target question: 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 1
This tell us that (p + n) is NOT DIVISIBLE by 5.
Since there's no information about (p-n), we can't determine the remainder when (p + n)(p - n) is divided by 15

Consider these two conflicting cases:
Case a: p = 5 and n = 1 (notice that the remainder when p+n is divided by 5 is 1). In this case, the remainder when is 9 when (p + n)(p - n) is divided by 15
Case b: p = 1 and n = 0 (notice that the remainder when p+n is divided by 5 is 1). In this case, the remainder when is 1 when (p + n)(p - n) is divided by 15
So, 2 of the numbers are less than 30
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The remainder when p - n is divided by 3 is 1
Here we have no information about p+n.
Consider these two conflicting cases:
Case a: p = 5 and n = 1 (notice that the remainder when p-n is divided by 3 is 1). In this case, the remainder when is 9 when (p + n)(p - n) is divided by 15
Case b: p = 1 and n = 0 (notice that the remainder when p-n is divided by 3 is 1). In this case, the remainder when is 1 when (p + n)(p - n) is divided by 15
So, 2 of the numbers are less than 30
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
IMPORTANT: Notice that I happened to use the same values for the counter-examples in each statement. This means that we can use the same values here to show that the COMBINED statements are not sufficient. That is...
Consider these two conflicting cases:
Case a: p = 5 and n = 1 (notice that both statements are satisfied). In this case, the remainder when is 9 when (p + n)(p - n) is divided by 15
Case b: p = 1 and n = 0 (notice that both statements are satisfied). In this case, the remainder when is 1 when (p + n)(p - n) is divided by 15
So, 2 of the numbers are less than 30
Since we cannot answer the target question with certainty, the COMBINED statements are NOT SUFFICIENT

Answer: E

ALTERNATIVELY, when examining the statements combined, we can use a nice rule that says:
If N divided by D, leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

Okay, onto the question . . .
Statement 1: Applying the above rule, some possible values of p+n are 6, 11, 16, 21, 26, etc.
Aside: you'll notice that I didn't include 1 as a possible value since we're told that p and n are positive integers, and we can't get a sum of 1 if both are positive

Statement 2: Applying the above rule, some possible values of p-n are 1, 4, 7, 10, 13, etc

Let's examine two cases with conflicting results.

case a: p+n = 11 and p-n = 1
Add the equations to get 2p = 12, which means p = 6 and n = 5 (perfect, we have positive integer values for p and n)
In this case, when (p + n)(p - n) is divided by 15, the remainder is 11

case b: p+n = 6 and p-n = 4
Add the equations to get 2p = 10, which means p = 5 and n = 1 (perfect, we have positive integer values for p and n)
In this case, when (p + n)(p - n) is divided by 15, the remainder is 9
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer = E

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Brent
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