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When 10 is divided by the positive integer n, the remainder
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15 Sep 2008, 14:33
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When 10 is divided by the positive integer n, the remainder is n4. Which of the following could be the value of n ? A) 3 B) 4 C) 7 D) 8 E) 12 OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/when10isd ... 68775.html
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Re: When 10 is divided by the positive integer n, the remainder
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18 Jun 2010, 01:58
jpr200012 wrote: When 10 is divided by the positive integer n, the remainder is n4. Which of the following could be the value of n?
A. 3 B. 4 C. 7 D. 8 E. 12
My strategy was to create lists below: n = 3, 4, 7, 8, 12 n4 = 1(becomes 9), 0, 3, 4, 8 n/10 = R? = 3, 4, 7, 8, 4
There is no match between n4 and n/10's R.
The solution uses 14 = ..., but I don't understand how they are using 14. Should the question have said a multiple of one of these numbers? Algebraic approach: THEORY:Positive integer \(a\) divided by positive integer \(d\) yields a reminder of \(r\) can always be expressed as \(a=qd+r\), where \(q\) is called a quotient and \(r\) is called a remainder, note here that \(0\leq{r}<d\) (remainder is nonnegative integer and always less than divisor). Original question says that when 10 is divided by the positive integer n, the remainder is n4, so \(10=nq+(n4)\) and also \(n4\geq{0}\) or \(n\geq{4}\) (remainder must be nonnegative). \(10=nq+n4\) > \(14=n(q+1)\) > as \(14=1*14=2*7\) and \(\geq{4}\) then > \(n\) can be 7 or 14. Answer: C. Hope it's clear.
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Re: When 10 is divided by the positive integer n, the remainder
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16 Sep 2008, 07:18
10=NK+N4; assume K=1
10=2N4
14=2N. N has to be a multiple of 7...
C it is..



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Re: When 10 is divided by the positive integer n, the remainder
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18 Jun 2010, 00:12
When 10 is divided by the positive integer n, the remainder is n4. Which of the following could be the value of n?
A. 3 B. 4 C. 7 D. 8 E. 12
My strategy was to create lists below: n = 3, 4, 7, 8, 12 n4 = 1(becomes 9), 0, 3, 4, 8 n/10 = R? = 3, 4, 7, 8, 4
There is no match between n4 and n/10's R.
The solution uses 14 = ..., but I don't understand how they are using 14. Should the question have said a multiple of one of these numbers?



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Re: When 10 is divided by the positive integer n, the remainder
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18 Jun 2010, 06:46
whiplash2411 wrote: It says that the remainder when you divide 10 by n is n4
This basically can be translated into the following statement algebraically:
\(10 = kn + (n4)\)
This is simplified as follows:
\(10 = kn + n 4 = n *(k+1)  4\)
Further simplifying:
\(10 + 4 = n*(k+1)
14 = n*(k+1)
7*2 = n*(k+1)\)
So n can be 7 or 2.
Only 7 is listed as an option here, so the answer is C. Hope this helps!
\(n\) cannot be 2 as in this case \(remainder =n4=2<0\) and remainder is always nonnegative (also notice that 10/2 has no remainder and n4=2, though n can also be 14 > 10=14*0+(144)). Hope it helps.
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Re: When 10 is divided by the positive integer n, the remainder
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16 Jul 2010, 04:22
Quote: remainder is always nonnegative Bunuel, I have to disagree with you on that: http://en.wikipedia.org/wiki/Remainder



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Re: When 10 is divided by the positive integer n, the remainder
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16 Jul 2010, 07:14
nonameee wrote: Quote: remainder is always nonnegative Bunuel, I have to disagree with you on that: http://en.wikipedia.org/wiki/RemainderThis has nothing to do with GMAT. GMAT Prep definition of the remainder:If \(a\) and \(d\) are positive integers, there exists unique integers \(q\) and \(r\), such that \(a = qd + r\) and \(0\leq{r}<d\). \(q\) is called a quotient and \(r\) is called a remainder. Also EVERY GMAT divisibility question will tell you in advance that any unknowns represent positive integers.So trust me: remainder is always nonnegative and less than divisor for GMAT  \(0\leq{r}<d\).
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Re: When 10 is divided by the positive integer n, the remainder
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Updated on: 06 Mar 2011, 16:27
I thought of the same , why cant a remainder be negative? I guess in some cases , as Bunel is suggesting we need to make an assumption that we are dealing with just positive integers. nonameee wrote: Quote: remainder is always nonnegative Bunuel, I have to disagree with you on that: http://en.wikipedia.org/wiki/Remainder
Originally posted by Spidy001 on 06 Mar 2011, 15:48.
Last edited by Spidy001 on 06 Mar 2011, 16:27, edited 1 time in total.



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Re: When 10 is divided by the positive integer n, the remainder
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06 Mar 2011, 15:55
Spidy001 wrote: I thought of the same , why cant a remainder be negative? I guess in some cases , as Bunel is suggesting we need an assumption that we are dealing with just positive integers. nonameee wrote: Quote: remainder is always nonnegative Bunuel, I have to disagree with you on that: http://en.wikipedia.org/wiki/RemainderIt's not an assumption. Remainder is a nonnegative by definition (at least on the GMAT).
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Re: When 10 is divided by the positive integer n, the remainder
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06 Mar 2011, 16:41
Bunuel, I know in this case we don't have to make any assumption, because the question clearly states these are two positive integers. i was referring more to scenarios like negative number division 25 /7 25 = 7(3)+(4) Here remainder is 4 which is negative. so lets say if question is like x,y are integers x/y . we cannot generalize and say remainder >=0 ,unless we assume that we are only talking about positive integers. nonameee wrote: Quote: remainder is always nonnegative Bunuel, I have to disagree with you on that: http://en.wikipedia.org/wiki/Remainder[/quote] It's not an assumption. Remainder is a nonnegative by definition (at least on the GMAT).[/quote]



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Re: When 10 is divided by the positive integer n, the remainder
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06 Mar 2011, 16:54
Spidy001 wrote: Bunuel,
I know in this case we don't have to make any assumption, because the question clearly states these are two positive integers.
i was referring more to scenarios like negative number division
25 /7
25 = 7(3)+(4)
Here remainder is 4 which is negative.
so lets say if question is like x,y are integers x/y . we cannot generalize and say remainder >=0 ,unless we assume that we are only talking about positive integers. Two things: 1. Every GMAT divisibility question will tell you in advance that any unknowns represent positive integers. 2. A remainder is a nonnegative integer by definition (at least on the GMAT).Anyway you are still wrong when calculating 25/7, it should be: 25=(4)*7+3, so remainder=3>0. TO SUMMARIZE, DON'T WORRY ABOUT NEGATIVE DIVIDENDS, DIVISORS OR REMAINDERS ON THE GMAT.
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Re: When 10 is divided by the positive integer n, the remainder
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09 Jan 2013, 04:02
Bunuel wrote: jpr200012 wrote: When 10 is divided by the positive integer n, the remainder is n4. Which of the following could be the value of n?
A. 3 B. 4 C. 7 D. 8 E. 12
My strategy was to create lists below: n = 3, 4, 7, 8, 12 n4 = 1(becomes 9), 0, 3, 4, 8 n/10 = R? = 3, 4, 7, 8, 4
There is no match between n4 and n/10's R.
The solution uses 14 = ..., but I don't understand how they are using 14. Should the question have said a multiple of one of these numbers? Algebraic approach: THEORY:Positive integer \(a\) divided by positive integer \(d\) yields a reminder of \(r\) can always be expressed as \(a=qd+r\), where \(q\) is called a quotient and \(r\) is called a remainder, note here that \(0\leq{r}<d\) (remainder is nonnegative integer and always less than divisor). Original question says that when 10 is divided by the positive integer n, the remainder is n4, so \(10=nq+(n4)\) and also \(n4\geq{0}\) or \(n\geq{4}\) (remainder must be nonnegative). \(10=nq+n4\) > \(14=n(q+1)\) > as \(14=1*14=2*7\) and \(\geq{4}\) then > \(n\) can be 7 or 14.Answer: C. Hope it's clear. So in this step are we substituting q=0,1 etc or is it something else?



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Re: When 10 is divided by the positive integer n, the remainder
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09 Jan 2013, 04:24
fozzzy wrote: Bunuel wrote: jpr200012 wrote: When 10 is divided by the positive integer n, the remainder is n4. Which of the following could be the value of n?
A. 3 B. 4 C. 7 D. 8 E. 12
My strategy was to create lists below: n = 3, 4, 7, 8, 12 n4 = 1(becomes 9), 0, 3, 4, 8 n/10 = R? = 3, 4, 7, 8, 4
There is no match between n4 and n/10's R.
The solution uses 14 = ..., but I don't understand how they are using 14. Should the question have said a multiple of one of these numbers? Algebraic approach: THEORY:Positive integer \(a\) divided by positive integer \(d\) yields a reminder of \(r\) can always be expressed as \(a=qd+r\), where \(q\) is called a quotient and \(r\) is called a remainder, note here that \(0\leq{r}<d\) (remainder is nonnegative integer and always less than divisor). Original question says that when 10 is divided by the positive integer n, the remainder is n4, so \(10=nq+(n4)\) and also \(n4\geq{0}\) or \(n\geq{4}\) (remainder must be nonnegative). \(10=nq+n4\) > \(14=n(q+1)\) > as \(14=1*14=2*7\) and \(\geq{4}\) then > \(n\) can be 7 or 14.Answer: C. Hope it's clear. So in this step are we substituting q=0,1 etc or is it something else? Not entirely so. From \(10=nq+n4\): Rearrange: \(14=nq+n\); Factor out n: \(14=n(q+1)\). So we have that the product of two positive integers (n and q+1) equals 14. 14 can be written as the product of two positive integers only in 2 way: 14=1*14 and 14=2*7. Now, since \(n\geq{4}\) then \(n\) can be 7 or 14. Hope it's clear.
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Re: When 10 is divided by the positive integer n, the remainder
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25 Sep 2013, 08:09
Bunuel wrote: jpr200012 wrote: When 10 is divided by the positive integer n, the remainder is n4. Which of the following could be the value of n?
A. 3 B. 4 C. 7 D. 8 E. 12
My strategy was to create lists below: n = 3, 4, 7, 8, 12 n4 = 1(becomes 9), 0, 3, 4, 8 n/10 = R? = 3, 4, 7, 8, 4
There is no match between n4 and n/10's R.
The solution uses 14 = ..., but I don't understand how they are using 14. Should the question have said a multiple of one of these numbers? Algebraic approach: THEORY:Positive integer \(a\) divided by positive integer \(d\) yields a reminder of \(r\) can always be expressed as \(a=qd+r\), where \(q\) is called a quotient and \(r\) is called a remainder, note here that \(0\leq{r}<d\) (remainder is nonnegative integer and always less than divisor). Original question says that when 10 is divided by the positive integer n, the remainder is n4, so \(10=nq+(n4)\) and also \(n4\geq{0}\) or \(n\geq{4}\) (remainder must be nonnegative). \(10=nq+n4\) > \(14=n(q+1)\) > as \(14=1*14=2*7\) and \(\geq{4}\) then > \(n\) can be 7 or 14. Answer: C. Hope it's clear. I got stuck when I got to 14=n(q+1)  so do we just completely ignore the 'q'? and why do we ignore the 'q'? can't q be something like 13 and 'n' becomes any random number? what am I missing here?



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Re: When 10 is divided by the positive integer n, the remainder
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25 Sep 2013, 08:32
bulletpoint wrote: Bunuel wrote: jpr200012 wrote: When 10 is divided by the positive integer n, the remainder is n4. Which of the following could be the value of n?
A. 3 B. 4 C. 7 D. 8 E. 12
My strategy was to create lists below: n = 3, 4, 7, 8, 12 n4 = 1(becomes 9), 0, 3, 4, 8 n/10 = R? = 3, 4, 7, 8, 4
There is no match between n4 and n/10's R.
The solution uses 14 = ..., but I don't understand how they are using 14. Should the question have said a multiple of one of these numbers? Algebraic approach: THEORY:Positive integer \(a\) divided by positive integer \(d\) yields a reminder of \(r\) can always be expressed as \(a=qd+r\), where \(q\) is called a quotient and \(r\) is called a remainder, note here that \(0\leq{r}<d\) (remainder is nonnegative integer and always less than divisor). Original question says that when 10 is divided by the positive integer n, the remainder is n4, so \(10=nq+(n4)\) and also \(n4\geq{0}\) or \(n\geq{4}\) (remainder must be nonnegative). \(10=nq+n4\) > \(14=n(q+1)\) > as \(14=1*14=2*7\) and \(\geq{4}\) then > \(n\) can be 7 or 14. Answer: C. Hope it's clear. I got stuck when I got to 14=n(q+1)  so do we just completely ignore the 'q'? and why do we ignore the 'q'? can't q be something like 13 and 'n' becomes any random number? what am I missing here? We don't ignore q, we are just not interested in it. q is a quotient, so is a nonnegative integer, thus we have 14=n(q+1)=integer*integer > both multiples are factors of 14. Does this make sense?
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Re: When 10 is divided by the positive integer n, the remainder
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Updated on: 26 Sep 2013, 03:00
Bunuel wrote: bulletpoint wrote: I got stuck when I got to 14=n(q+1)  so do we just completely ignore the 'q'? and why do we ignore the 'q'? can't q be something like 13 and 'n' becomes any random number? what am I missing here?
We don't ignore q, we are just not interested in it. q is a quotient, so is a nonnegative integer, thus we have 14=n(q+1)=integer*integer > both multiples are factors of 14. Does this make sense? why do both 'n' and '(q+1)' have to be factors of 14? if 'q+1' is a factor of 14, then 'n' need not be a factor of 14 for the equation 14=n(q+1) to be true, right? or is it that for questions of these types  since we are only interested in what 'n' is  we just completely ignore the '(q+1)' part? EDIT: Just took a look at what you said again and I think I get it. Please correct me if I'm wrong: 14=n(q+1) means 'n' OR '(q+1)' can equal 1,2,7,14 to make the equation true, and since 'n' has to be greater or equal to 4 because remainder must be nonnegative, it can only be true that 'n' equals 7 or 14, and because the answer only has 7, this would be the correct answer.
Originally posted by bulletpoint on 26 Sep 2013, 02:56.
Last edited by bulletpoint on 26 Sep 2013, 03:00, edited 1 time in total.



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Re: When 10 is divided by the positive integer n, the remainder
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26 Sep 2013, 02:59
bulletpoint wrote: Bunuel wrote: bulletpoint wrote: I got stuck when I got to 14=n(q+1)  so do we just completely ignore the 'q'? and why do we ignore the 'q'? can't q be something like 13 and 'n' becomes any random number? what am I missing here?
We don't ignore q, we are just not interested in it. q is a quotient, so is a nonnegative integer, thus we have 14=n(q+1)=integer*integer > both multiples are factors of 14. Does this make sense? why do both 'n' and '(q+1)' have to be factors of 14? if 'q+1' is a factor of 14, then 'n' need not be a factor of 14 for the equation 14=n(q+1) to be true, right? or is it that for questions of these types  since we are only interested in what 'n' is  we just completely ignore the '(q+1)' part? Again we do NOT ignore q+1. Next, 14 = n(q+1) = integer*integer: 14/n = q+1 = integer > n is a factor of 14. 14/(q+1) = n = integer > q+1 is a factor of 14.
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Re: When 10 is divided by the positive integer n, the remainder
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23 Oct 2013, 18:42
Bunuel wrote: jpr200012 wrote: When 10 is divided by the positive integer n, the remainder is n4. Which of the following could be the value of n?
A. 3 B. 4 C. 7 D. 8 E. 12
My strategy was to create lists below: n = 3, 4, 7, 8, 12 n4 = 1(becomes 9), 0, 3, 4, 8 n/10 = R? = 3, 4, 7, 8, 4
There is no match between n4 and n/10's R.
The solution uses 14 = ..., but I don't understand how they are using 14. Should the question have said a multiple of one of these numbers? Algebraic approach: THEORY:Positive integer \(a\) divided by positive integer \(d\) yields a reminder of \(r\) can always be expressed as \(a=qd+r\), where \(q\) is called a quotient and \(r\) is called a remainder, note here that \(0\leq{r}<d\) (remainder is nonnegative integer and always less than divisor). Original question says that when 10 is divided by the positive integer n, the remainder is n4, so \(10=nq+(n4)\) and also \(n4\geq{0}\) or \(n\geq{4}\) (remainder must be nonnegative). \(10=nq+n4\) > \(14=n(q+1)\) > as \(14=1*14=2*7\) and \(\geq{4}\) then > \(n\) can be 7 or 14.Answer: C. Hope it's clear. could you clarify the highlighted portion? is n being 7 because 14=2*7?



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Re: When 10 is divided by the positive integer n, the remainder
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24 Oct 2013, 00:20
AccipiterQ wrote: Bunuel wrote: jpr200012 wrote: When 10 is divided by the positive integer n, the remainder is n4. Which of the following could be the value of n?
A. 3 B. 4 C. 7 D. 8 E. 12
My strategy was to create lists below: n = 3, 4, 7, 8, 12 n4 = 1(becomes 9), 0, 3, 4, 8 n/10 = R? = 3, 4, 7, 8, 4
There is no match between n4 and n/10's R.
The solution uses 14 = ..., but I don't understand how they are using 14. Should the question have said a multiple of one of these numbers? Algebraic approach: THEORY:Positive integer \(a\) divided by positive integer \(d\) yields a reminder of \(r\) can always be expressed as \(a=qd+r\), where \(q\) is called a quotient and \(r\) is called a remainder, note here that \(0\leq{r}<d\) (remainder is nonnegative integer and always less than divisor). Original question says that when 10 is divided by the positive integer n, the remainder is n4, so \(10=nq+(n4)\) and also \(n4\geq{0}\) or \(n\geq{4}\) (remainder must be nonnegative). \(10=nq+n4\) > \(14=n(q+1)\) > as \(14=1*14=2*7\) and \(\geq{4}\) then > \(n\) can be 7 or 14.Answer: C. Hope it's clear. could you clarify the highlighted portion? is n being 7 because 14=2*7? Yes, we know that \(n\geq{4}\) and \(14=n*(positive \ integer)\). Now, \(14=1*14=2*7\), thus \(n\) can be 7 or 14. Hope it's clear.
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Re: When 10 is divided by the positive integer n, the remainder
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05 May 2015, 06:26
Here's a slightly trickier question that is good for further practice on this OG QR2 question: http://gmatclub.com/forum/when81isdividedbythecubeofpositiveintegerztheremainderis197386.html#p1522792Wish you all the best!
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Re: When 10 is divided by the positive integer n, the remainder
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