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# When 10 is divided by the positive integer n, the remainder

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Joined: 10 Mar 2008
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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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Question Stats:

89% (01:12) correct 11% (01:24) wrong based on 519 sessions

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When 10 is divided by the positive integer n, the remainder is n-4. 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/when-10-is-d ... 68775.html
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Joined: 02 Sep 2009
Posts: 58434
Re: When 10 is divided by the positive integer n, the remainder  [#permalink]

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18 Jun 2010, 01:58
6
4
jpr200012 wrote:
When 10 is divided by the positive integer n, the remainder is n-4. 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
n-4 = -1(becomes 9), 0, 3, 4, 8
n/10 = R? = 3, 4, 7, 8, 4

There is no match between n-4 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 non-negative integer and always less than divisor).

Original question says that when 10 is divided by the positive integer n, the remainder is n-4, so $$10=nq+(n-4)$$ and also $$n-4\geq{0}$$ or $$n\geq{4}$$ (remainder must be non-negative).

$$10=nq+n-4$$ --> $$14=n(q+1)$$ --> as $$14=1*14=2*7$$ and $$\geq{4}$$ 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  [#permalink]

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16 Sep 2008, 07:18
3
1
10=NK+N-4; assume K=1

10=2N-4

14=2N. N has to be a multiple of 7...

C it is..
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Joined: 30 May 2010
Posts: 165
Re: When 10 is divided by the positive integer n, the remainder  [#permalink]

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18 Jun 2010, 00:12
When 10 is divided by the positive integer n, the remainder is n-4. 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
n-4 = -1(becomes 9), 0, 3, 4, 8
n/10 = R? = 3, 4, 7, 8, 4

There is no match between n-4 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?
Math Expert
Joined: 02 Sep 2009
Posts: 58434
Re: When 10 is divided by the positive integer n, the remainder  [#permalink]

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18 Jun 2010, 06:46
whiplash2411 wrote:
It says that the remainder when you divide 10 by n is n-4

This basically can be translated into the following statement algebraically:

$$10 = kn + (n-4)$$

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 $$remainder =n-4=-2<0$$ and remainder is always non-negative (also notice that 10/2 has no remainder and n-4=-2, though n can also be 14 --> 10=14*0+(14-4)).

Hope it helps.
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Posts: 506
Re: When 10 is divided by the positive integer n, the remainder  [#permalink]

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16 Jul 2010, 04:22
Quote:
remainder is always non-negative

Bunuel, I have to disagree with you on that:

http://en.wikipedia.org/wiki/Remainder
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Posts: 58434
Re: When 10 is divided by the positive integer n, the remainder  [#permalink]

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16 Jul 2010, 07:14
1
nonameee wrote:
Quote:
remainder is always non-negative

Bunuel, I have to disagree with you on that:

http://en.wikipedia.org/wiki/Remainder

This 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 non-negative 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  [#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 non-negative

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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Posts: 58434
Re: When 10 is divided by the positive integer n, the remainder  [#permalink]

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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 non-negative

Bunuel, I have to disagree with you on that:

http://en.wikipedia.org/wiki/Remainder

It's not an assumption.

Remainder is a non-negative by definition (at least on the GMAT).
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Re: When 10 is divided by the positive integer n, the remainder  [#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 non-negative

Bunuel, I have to disagree with you on that:

http://en.wikipedia.org/wiki/Remainder
[/quote]

It's not an assumption.

Remainder is a non-negative by definition (at least on the GMAT).[/quote]
Math Expert
Joined: 02 Sep 2009
Posts: 58434
Re: When 10 is divided by the positive integer n, the remainder  [#permalink]

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06 Mar 2011, 16:54
1
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 non-negative 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  [#permalink]

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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 n-4. 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
n-4 = -1(becomes 9), 0, 3, 4, 8
n/10 = R? = 3, 4, 7, 8, 4

There is no match between n-4 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 non-negative integer and always less than divisor).

Original question says that when 10 is divided by the positive integer n, the remainder is n-4, so $$10=nq+(n-4)$$ and also $$n-4\geq{0}$$ or $$n\geq{4}$$ (remainder must be non-negative).

$$10=nq+n-4$$ --> $$14=n(q+1)$$ --> as $$14=1*14=2*7$$ and $$\geq{4}$$ then --> $$n$$ can be 7 or 14.

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  [#permalink]

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09 Jan 2013, 04:24
1
fozzzy wrote:
Bunuel wrote:
jpr200012 wrote:
When 10 is divided by the positive integer n, the remainder is n-4. 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
n-4 = -1(becomes 9), 0, 3, 4, 8
n/10 = R? = 3, 4, 7, 8, 4

There is no match between n-4 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 non-negative integer and always less than divisor).

Original question says that when 10 is divided by the positive integer n, the remainder is n-4, so $$10=nq+(n-4)$$ and also $$n-4\geq{0}$$ or $$n\geq{4}$$ (remainder must be non-negative).

$$10=nq+n-4$$ --> $$14=n(q+1)$$ --> as $$14=1*14=2*7$$ and $$\geq{4}$$ then --> $$n$$ can be 7 or 14.

Hope it's clear.

So in this step are we substituting q=0,1 etc or is it something else?

Not entirely so.

From $$10=nq+n-4$$:

Re-arrange: $$14=nq+n$$;
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  [#permalink]

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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 n-4. 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
n-4 = -1(becomes 9), 0, 3, 4, 8
n/10 = R? = 3, 4, 7, 8, 4

There is no match between n-4 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 non-negative integer and always less than divisor).

Original question says that when 10 is divided by the positive integer n, the remainder is n-4, so $$10=nq+(n-4)$$ and also $$n-4\geq{0}$$ or $$n\geq{4}$$ (remainder must be non-negative).

$$10=nq+n-4$$ --> $$14=n(q+1)$$ --> as $$14=1*14=2*7$$ and $$\geq{4}$$ then --> $$n$$ can be 7 or 14.

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  [#permalink]

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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 n-4. 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
n-4 = -1(becomes 9), 0, 3, 4, 8
n/10 = R? = 3, 4, 7, 8, 4

There is no match between n-4 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 non-negative integer and always less than divisor).

Original question says that when 10 is divided by the positive integer n, the remainder is n-4, so $$10=nq+(n-4)$$ and also $$n-4\geq{0}$$ or $$n\geq{4}$$ (remainder must be non-negative).

$$10=nq+n-4$$ --> $$14=n(q+1)$$ --> as $$14=1*14=2*7$$ and $$\geq{4}$$ then --> $$n$$ can be 7 or 14.

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 non-negative 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  [#permalink]

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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 non-negative 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 non-negative, 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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Posts: 58434
Re: When 10 is divided by the positive integer n, the remainder  [#permalink]

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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 non-negative 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  [#permalink]

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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 n-4. 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
n-4 = -1(becomes 9), 0, 3, 4, 8
n/10 = R? = 3, 4, 7, 8, 4

There is no match between n-4 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 non-negative integer and always less than divisor).

Original question says that when 10 is divided by the positive integer n, the remainder is n-4, so $$10=nq+(n-4)$$ and also $$n-4\geq{0}$$ or $$n\geq{4}$$ (remainder must be non-negative).

$$10=nq+n-4$$ --> $$14=n(q+1)$$ --> as $$14=1*14=2*7$$ and $$\geq{4}$$ then --> $$n$$ can be 7 or 14.

Hope it's clear.

could you clarify the highlighted portion? is n being 7 because 14=2*7?
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Posts: 58434
Re: When 10 is divided by the positive integer n, the remainder  [#permalink]

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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 n-4. 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
n-4 = -1(becomes 9), 0, 3, 4, 8
n/10 = R? = 3, 4, 7, 8, 4

There is no match between n-4 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 non-negative integer and always less than divisor).

Original question says that when 10 is divided by the positive integer n, the remainder is n-4, so $$10=nq+(n-4)$$ and also $$n-4\geq{0}$$ or $$n\geq{4}$$ (remainder must be non-negative).

$$10=nq+n-4$$ --> $$14=n(q+1)$$ --> as $$14=1*14=2*7$$ and $$\geq{4}$$ then --> $$n$$ can be 7 or 14.

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  [#permalink]

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05 May 2015, 06:26
Re: When 10 is divided by the positive integer n, the remainder   [#permalink] 05 May 2015, 06:26

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