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When 10 is divided by the positive integer n, the remainder [#permalink]
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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
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Re: Number properties question from QR 2nd edition PS 164 [#permalink]
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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: PS: Remainder Theory [#permalink]
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15 Sep 2008, 17:08
back solving is easier for this one.
10/7 gives a remainder of 3. n4 = 74 = 3



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Re: PS: Remainder Theory [#permalink]
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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: PS: Remainder Theory [#permalink]
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17 Sep 2008, 07:04
vksunder wrote: fresinha12  I did the same way as you had described. But is it safe to assume that K=1? well, since we can never have the denominator to be zero, otherwise the fraction will be undefined. so it makes sense to start off with k=1. If that doesn't work, then you just have to keep increasing the value of k until you can match your answer with the correct answer choice.



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Re: PS: Remainder Theory [#permalink]
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18 Sep 2008, 00:18
I approached as follows.
10 = nx + n4 for x = 0,1,2,3,4...... or, n(x+1) = 14 or, n = 14/(x+1)
For x = 0, n = 14, for x = 1, n = 7, x cannot be 2,3,4,5. For x = 6, n = 2. x cannot be greater than 6.
Hence, possible values of n are 14, 7, 2. Answer choice has 7. Hence, 7 is the answer.



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Number properties question from QR 2nd edition PS 164 [#permalink]
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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: Number properties question from QR 2nd edition PS 164 [#permalink]
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18 Jun 2010, 00:56
As per my approach, it is easy to reach the solution by going thorough each one of the options. You can eliminate 12,8,4 and 3 at one look. Then you just need to check for 7. It took me less than 1 minute to get to the answer. So that should be fine I guess.



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Re: Number properties question from QR 2nd edition PS 164 [#permalink]
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18 Jun 2010, 06:29
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! 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?



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Re: Number properties question from QR 2nd edition PS 164 [#permalink]
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18 Jun 2010, 06:46



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Re: Number properties question from QR 2nd edition PS 164 [#permalink]
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18 Jun 2010, 06:52
Oh, yeah, that's right. I just saw the 7 and 2, and looked at the answer choices and chose 7. Thanks, Bunuel. Your explanation will come in handy in case both 2 and 7 were listed as answer choices!



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Re: Number properties question from QR 2nd edition PS 164 [#permalink]
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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: Number properties question from QR 2nd edition PS 164 [#permalink]
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16 Jul 2010, 07:14



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Re: Number properties question from QR 2nd edition PS 164 [#permalink]
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16 Jul 2010, 14:00
Thanks for clarification. But you can use that property (negative remainder) to solve remainder problems (as it has been done in several posts).



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Re: QR. 164 Remainder [#permalink]
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05 Mar 2011, 01:24
If division by n leaves reminder. Then i.e. Dividend  Remainder is a multiple of divider. Here 10 (n4) must be a multiple of n. Or Is [10  (n4)] / n = integer?Now plug in the values of n from the options. A  n4 will give negative remainder. Illogical B  (100)/4 is not integer C  (103)/7 is integer D  (104)/8 is not integer E  (108)/12 is not integer Answer C. Baten80 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



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Re: QR. 164 Remainder [#permalink]
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05 Mar 2011, 01:44
\(10=nQ+n4\) where Q is the quotient \(n(Q+1)=14\), where n and Q are both integers. Factors of 14; n*(Q+1) 1*14; n=1, Q=13; Not possible because 1 won't leave any remainder with 10 2*7; n=2, Q=6; Not possible because 2 won't leave any remainder with 10 7*2; n=7, Q=1; Possible 14*1; n=14, Q=0; Possible So; n can be 7 or 14. Ans: "C"
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Re: Number properties question from QR 2nd edition PS 164 [#permalink]
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06 Mar 2011, 15:04
Nice explanation there Bunuel. 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). \(10=nq+n4\) > \(14=n(q+1)\) > \(n\) is an factor of 14 and \(\geq{4}\) > \(n\) can be 7 or 14. Answer: C. Hope it's clear.



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Re: Number properties question from QR 2nd edition PS 164 [#permalink]
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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: Number properties question from QR 2nd edition PS 164 [#permalink]
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06 Mar 2011, 15:55



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Re: Number properties question from QR 2nd edition PS 164 [#permalink]
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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]




Re: Number properties question from QR 2nd edition PS 164
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