If p and n are positive integers and p>n , what is the remainder when (p*p)-(n*n) is divided by 15?
A. remainder when p+n is divided by 5 is 1.
B. remainder when p-n is divided by 3 is 1.
GMAT Prep question
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Question : Remainder when (p^2 - n^2) is divided by 15anksm22 wrote:If p and n are positive integers and p>n , what is the remainder when (p*p)-(n*n) is divided by 15?
A. remainder when p+n is divided by 5 is 1.
B. remainder when p-n is divided by 3 is 1.
Question : Remainder when (p-n)x(p+n) is divided by (3x5)
Statement 1) remainder when p+n is divided by 5 is 1.
It only answers half of the question because the remainder when the entire expression (p+n)(p-n) is divided by 15 is unknown, information about p-n is unknown and divisor 3 is absent in the statement 1
Insufficient
Statement 2) remainder when p-n is divided by 3 is 1.
It only answers half of the question because the remainder when the entire expression (p+n)(p-n) is divided by 15 is unknown, p+n is unknown and divisor 5 is absent in the statement 2
Insufficient
Combining the two statements
(p+n) can be 6, 11, 16, 21, 26, 31, 36, ... etc
(p-n) can be 4, 7, 10, 13, 16, 19, 22, ... etc
i.e. (p+n)(p-n) can be 6x4 = 24 leaving remainder 9 when divided by 15
or (p+n)(p-n) can be 11x7 = 77 leaving remainder 2 when divided by 15
or (p+n)(p-n) can be 16x10 = 160 leaving remainder 10 when divided by 15
Inconsistent answer therefore INSUFFICIENT
Answer: Option E
Last edited by GMATinsight on Sun Jul 20, 2014 2:30 am, edited 1 time in total.
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Target question: What is the remainder when p² - n² is divided by 15If 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.
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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Question rephrased: What is the remainder when (p+n)(p-n) is divided by 15?anksm22 wrote: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.
Statement 1:
p+n = 5a + 1, where a is an integer such that a≥0.
No information about p-n.
INSUFFICIENT.
Statement 2:
p-n = 3b + 1, where b is an integer such that b≥0.
No information about p+n.
INSUFFICIENT.
Statements combined:
(p+n)(p-n) = (5a + 1)(3b + 1).
Note the following constraints:
Since n>0, p+n > p-n.
Since (p+n) + (p-n) = 2p = even, p+n and p-n must either both be EVEN or both be ODD.
Case 1: a=1 and b=1
Here, (p+n)(p-n) = (5a + 1)(3b + 1) = (5*1 + 1)(3*1 + 1) = 6*4 = 24.
In this case, 24/15 = 1 R9.
Case 2: a=2 and b=2
Here, (p+n)(p-n) = (5a + 1)(3b + 1) = (5*2 + 1)(3*2 + 1) = 11*7 = 77.
In this case, 77/15 = 5 R2.
Since R can be different values, INSUFFICIENT.
The correct answer is E.
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