Hi,
Need some assistance with the following question.
Question
If "n" is a positive integer and "r" is the remainder when (n-1)(n+1) is divided by 24.
What is the value of "r"?
1. n is not divisible by 2
2. n is not divisible by 3
Thanks
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Remainders
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GMATGuruNY
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Statement 1: n is not divisible by 2binaras wrote: If "n" is a positive integer and "r" is the remainder when (n-1)(n+1) is divided by 24.
What is the value of "r"?
1. n is not divisible by 2
2. n is not divisible by 3
Options for n:
1, 3, 5, 7, 9, 11, 13...
If n=1, then dividing (n+1)(n-1) by 24 yields the following:
(1+1)(1-1) / 24 = 0/24 = 0 R0.
If n=3, then dividing (n+1)(n-1) by 24 yields the following:
(3+1)(3-1) / 24 = 8/24 = 0 R8.
Since R can be different values, INSUFFICIENT.
Statement 2: n is not divisible by 3
Options for n:
1, 2, 4, 5, 7, 8...
If n=1, then dividing (n+1)(n-1) by 24 yields the following:
(1+1)(1-1) / 24 = 0/24 = 0 R0.
If n=2, then dividing (n+1)(n-1) by 24 yields the following:
(2+1)(2-1) / 24 = 3/24 = 0 R3.
Since R can be different values, INSUFFICIENT.
Statements combined:
Options for n:
1, 5, 7, 11...
If n=1, then dividing (n+1)(n-1) by 24 yields the following:
(1+1)(1-1) / 24 = 0/24 = 0 R0.
If n=5, then dividing (n+1)(n-1) by 24 yields the following:
(5+1)(5-1) / 24 = 24/24 = 1 R0.
If n=7, then dividing (n+1)(n-1) by 24 yields the following:
(7+1)(7-1) / 24 = 48/24 = 2 R0.
If n=11, then dividing (n+1)(n-1) by 24 yields the following:
(11+1)(11-1) / 24 = 120/24 = 5 R0.
In every case, R=0.
SUFFICIENT.
The correct answer is C.
Alternate approach:
Statement 1: 2 is not a factor of n.
Thus, n = odd.
Thus, (n-1)(n+1) = the product of two consecutive even integers.
Of every two consecutive even integers, exactly one is a multiple of 4.
Thus, the product of 2 consecutive even integers = the product of an even integer and a multiple of 4 = a multiple of 8.
Since a multiple of 8 can be a multiple of 24 (in which case r=0) or not be a multiple of 24 (in which case r≠0), INSUFFICIENT.
Statement 2: 3 is not a factor of n
Since one of every 3 consecutive integers is a multiple of 3, and n is not a multiple of 3, either (n-1) or (n+1) must be a multiple of 3.
Thus, (n-1)(n+1) = a multiple of 3.
If (n-1)(n+1) is also a multiple of 8, then (n-1)(n+1) = a multiple of 24, in which case r=0.
If (n-1)(n+1) is not a multiple of 8, then (n-1)(n+1) ≠a multiple of 24, in which case r≠0.
INSUFFICIENT.
Statements 1 and 2 combined:
Since (n-1)(n+1) = a multiple of 8, and either n-1 or n+1 must be a multiple of 3, (n-1)(n+1) = a multiple of 24.
When a multiple of 24 is divided by 24, r=0.
SUFFICIENT.
The correct answer is C.
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ceilidh.erickson
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The structure (n - 1)(n + 1) is code for either consecutive even integers or consecutive odd integers. (If n is even, it'll be the integer below times the integer above, and thus the product of two consecutive odds. If n is odd, it will be the productive of 2 consecutive evens).
scenario 1: If (n - 1) and (n + 1) are consecutive odd integers, the product will never be divisible by 24.
3*5, 5*7, 9*11, etc.
In these 3 examples, we get 3 different remainder. So if n is even, we'd need a value for n to know the remainder.
scenario 2: If (n - 1) and (n + 1) are consecutive even integers, the product will be divisible by 24 if one of the two terms is divisible by 3:
4*6, 6*8, 10*12, 18*20, etc.
If n is NOT divisible by 3, then either the integer above it or the integer below it MUST be divisible by 3.
scenario 3: If (n - 1) and (n + 1) are consecutive even integers, the product will NOT be divisible by 24 if neither term is divisible by 3:
2*4, 8*10, 14*16, etc.
But, with any of these products, the remainder will be 8.
If we know that n IS divisible by 3, then the integers above and below it can't be - we would have a remainder of 8.
Target question: Is n odd (making n - 1 and n + 1 both even), and do we know whether it's divisible by 3?
1) n is not divisible by 2.
This eliminates scenario 1, and tells us that we have the product of consecutive even integers. However, we could still have scenario 2 or 3, giving us a remainder of 0 or 8. Insufficient.
2) n is not divisible by 3.
This doesn't tell us whether n is even or odd. Insufficient.
1&2) If n is both odd and NOT a multiple of 3, then (n - 1) and (n + 1) are even, and one of the two of them must be divisible by 3. Thus, the product must be divisible by 24. Sufficient!
The answer is C.
scenario 1: If (n - 1) and (n + 1) are consecutive odd integers, the product will never be divisible by 24.
3*5, 5*7, 9*11, etc.
In these 3 examples, we get 3 different remainder. So if n is even, we'd need a value for n to know the remainder.
scenario 2: If (n - 1) and (n + 1) are consecutive even integers, the product will be divisible by 24 if one of the two terms is divisible by 3:
4*6, 6*8, 10*12, 18*20, etc.
If n is NOT divisible by 3, then either the integer above it or the integer below it MUST be divisible by 3.
scenario 3: If (n - 1) and (n + 1) are consecutive even integers, the product will NOT be divisible by 24 if neither term is divisible by 3:
2*4, 8*10, 14*16, etc.
But, with any of these products, the remainder will be 8.
If we know that n IS divisible by 3, then the integers above and below it can't be - we would have a remainder of 8.
Target question: Is n odd (making n - 1 and n + 1 both even), and do we know whether it's divisible by 3?
1) n is not divisible by 2.
This eliminates scenario 1, and tells us that we have the product of consecutive even integers. However, we could still have scenario 2 or 3, giving us a remainder of 0 or 8. Insufficient.
2) n is not divisible by 3.
This doesn't tell us whether n is even or odd. Insufficient.
1&2) If n is both odd and NOT a multiple of 3, then (n - 1) and (n + 1) are even, and one of the two of them must be divisible by 3. Thus, the product must be divisible by 24. Sufficient!
The answer is C.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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ceilidh.erickson
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Here are some examples of similarly structured questions that test consecutive products:
https://www.beatthegmat.com/is-x-x-2-x-4 ... tml#718646
https://www.beatthegmat.com/totaly-lost- ... tml#716315
https://www.beatthegmat.com/is-x-x-2-x-4 ... tml#718646
https://www.beatthegmat.com/totaly-lost- ... tml#716315
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
GMAT/MBA Expert
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Jeff@TargetTestPrep
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Solution:binaras wrote:Hi,
Need some assistance with the following question.
Question
If "n" is a positive integer and "r" is the remainder when (n-1)(n+1) is divided by 24.
What is the value of "r"?
1. n is not divisible by 2
2. n is not divisible by 3
Thanks
This is a great question for choosing convenient numbers for the variables. When setting up this question, we can also use the difference of squares since we know that (n-1)(n+1) is equal to n^2 - 1. Setting up the question into a mathematical expression, we can say:
(n^2 - 1)/24, Remainder = ?
Statement One Alone:
n is not divisible by 2
This tells us that n is an odd number. Thus, we can try a few different odd numbers, starting with 1. When n = 1, n^2 - 1 = 0. The remainder of 0/24 is 0.
Now let's use n = 3.
When n = 3, n^2 - 1 = 8. The remainder of 8/24 is 8.
Because we have found two different remainders (or values for r), statement one alone is insufficient. We can eliminate answers A and D.
Statement Two Alone:
n is not divisible by 3
This tells us that n cannot be a multiple of 3. Since we have already tested "1," a non-multiple of 3, in statement one, we can again use that value in statement two.
When n = 1, n^2 - 1 = 0. The remainder of 0/24 is 0.
Now let's use n = 2.
When n = 2, n^2 - 1 = 3. The remainder of 3/24 is 3.
Again we again see that we have found two different remainders (or values for r); statement two alone is insufficient. We can eliminate answer B.
Statements One and Two Together:
We know that n cannot be even, nor can it be a multiple of 3. We know that 1 is a possible value for n and that when n = 1, 1^2 - 1 = 0. The remainder of 0/24 is 0.
Next we can use n = 5 (we can't use 2, 3 or 4 since they are either even or a multiple of 3).
When n = 5, n^2 - 1 = 24. The remainder of 24/24 is 0.
Now it appears we have found a pattern that, when we continue to plug in values, fulfills statements one and two: we will get a remainder of zero; however, let's test one more value of n for good measure.
Let's use n = 7.
When n = 7, n^2 - 1 = 48. The remainder of 48/24 is 0.
The answer is C
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