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# When 10 is divided by the positive integer n, the remainder is n - 4.

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Math Expert
Joined: 02 Sep 2009
Posts: 58410
When 10 is divided by the positive integer n, the remainder is n - 4.  [#permalink]

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14 Mar 2014, 03:18
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5% (low)

Question Stats:

87% (01:20) correct 13% (01:33) wrong based on 777 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

Problem Solving
Question: 164
Category: Algebra Properties of numbers
Page: 84
Difficulty: 600

The Official Guide For GMAT® Quantitative Review, 2ND Edition

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Math Expert
Joined: 02 Sep 2009
Posts: 58410
Re: When 10 is divided by the positive integer n, the remainder is n - 4.  [#permalink]

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16 Mar 2014, 05:55
1
6
SOLUTION

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

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.

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Joined: 26 Feb 2015
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Re: When 10 is divided by the positive integer n, the remainder is n - 4.  [#permalink]

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14 Mar 2015, 07:08
Bunuel wrote:
SOLUTION

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

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.

I simply did it this way:

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

And then I just plugged in values for n. So:
$$10(0) + (0-4) = -4$$
$$10(1) + (1-4) = 7$$ <--my answer

Is this correct reasoning, or simply a lucky coincidence?
Math Expert
Joined: 02 Aug 2009
Posts: 8005
Re: When 10 is divided by the positive integer n, the remainder is n - 4.  [#permalink]

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14 Mar 2015, 07:39
erikvm wrote:
Bunuel wrote:
SOLUTION

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

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.

I simply did it this way:

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

And then I just plugged in values for n. So:
$$10(0) + (0-4) = -4$$
$$10(1) + (1-4) = 7$$ <--my answer

Is this correct reasoning, or simply a lucky coincidence?

hi erikvm,
here it is a lucky coincidence..
here you have found the value of y and not n...

$$10(1) + (1-4) = 7$$... here n=1 and y=7.... but the question asks us the value of n...
the equation that you have formed is wrong ..
it says when 10 is divided by the positive integer n, the remainder is n-4.. so the equation will be 10=xn+n-4..

what we have to find is that " when 10 is divided by n, the remainder is n-4"
now substitue the values
(A) 3.... 10/3 rem=1, which is not equal to n-4=3-4=-1.... not correct
(B) 4.... 10/4 rem=2, which is not equal to n-4=4-4=0.... not correct
(C) 7.... 10/7 rem=3, which is equal to n-4=7-4=3.... correct
(D) 8.... 10/8 rem=2, which is not equal to n-4=8-4=2.... not correct
(E) 12... 10/12 rem=10, which is not equal to n-4=10-4=6.... not correct

hope it is clear..
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When 10 is divided by the positive integer n, the remainder is n - 4.  [#permalink]

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13 Apr 2019, 15:54
1
Hi,

Here are my two cents for this question. This solution is same as all others but want to share one important aspect of thinking in case of remainders.

My solution would appear little lengthy, but if you can visualize certain things about remainders, we could solve this question this question in about 33 Secs.

Here we are told that when $$\frac{10}{n}$$ we have remainder in the form (n-4)

So we can write this $$\frac{10}{n}$$ as 10 = nk +(n-4) ------(1)

where 0$$\leq{reminder}$$<n ; where 0$$\leq{n-4}$$<n ( This is the first hint toward rejecting incorrect choice A, B )

which says 4$$\leq{n}$$

here we can have two cases
Case 1: if n>10
then nk =0 and we can re-write the equation (1) as .
10 = n-4
where n =14
This is our second hint for rejecting Answer Choice "E" Consider a case where n = 12 so we have $$\frac{10}{12}$$= 12*0+ 10) here remainder is 10 . But using the equation given to us in the question stem that remainder is n-4 , which in this case would be 12-4 =8 So we can reject answer choice E

Case 2:
If n< 10,
then we have k is a positive integer.
$$\frac{n}{10}$$ = nk +(n-4) ------(1)
and simplifying we get
10+4= nk+n
14=n(k+1)
so we have
14= 7(1+1)
so we have n=7

Hence form given choices we have answer to the question which is choice "C"
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When 10 is divided by the positive integer n, the remainder is n - 4.   [#permalink] 13 Apr 2019, 15:54
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