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Math Expert V
Joined: 02 Sep 2009
Posts: 62676
When 10 is divided by the positive integer n, the remainder is n - 4.  [#permalink]

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Difficulty:   5% (low)

Question Stats: 87% (01:20) correct 13% (01:36) wrong based on 903 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 V
Joined: 02 Sep 2009
Posts: 62676
Re: When 10 is divided by the positive integer n, the remainder is n - 4.  [#permalink]

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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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Posts: 109
Re: When 10 is divided by the positive integer n, the remainder is n - 4.  [#permalink]

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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 V
Joined: 02 Aug 2009
Posts: 8331
Re: When 10 is divided by the positive integer n, the remainder is n - 4.  [#permalink]

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

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From the problem we know that: 10 = N*K + (N-4)
to simplify: 14 = N*K + N => K = (14 - N)/N and if we insert answers here the only one possible would be 7.
Intern  B
Joined: 12 Feb 2020
Posts: 1
Re: When 10 is divided by the positive integer n, the remainder is n - 4.  [#permalink]

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This could be solved easily by testing out the options.

answer is c as the remainder will be n-4 when n =7.
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GMAT 1: 780 Q51 V45
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Re: When 10 is divided by the positive integer n, the remainder is n - 4.  [#permalink]

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Bunuel 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

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

We can always plug in one by one and test which works.
(A) The remainder is 3 - 4 = -1 which doesn't make sense.
(B) 10/4 gives a remainder of 2, n - 4 = 4 - 4 = 0 however so not B.
(C) 10/7 gives a remainder of 3. n - 4 = 7 - 4 = 3. Bingo.

Ans: C
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Re: When 10 is divided by the positive integer n, the remainder is n - 4.  [#permalink]

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Bunuel 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

10/n=q+(n-4)→
n=14/(q+1)
only 7 works
C
GMAT Club Legend  V
Joined: 11 Sep 2015
Posts: 4605
GMAT 1: 770 Q49 V46
When 10 is divided by the positive integer n, the remainder is n - 4.  [#permalink]

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Top Contributor
Bunuel 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

APPROACH #1: I'd say that the fastest approach is to simply test answer choices

(A) 3
The question tells us that we get a remainder of n - 4
So, if n = 3, then the remainder = 3 - 4 = -1, which makes no sense (the remainder must always be greater than or equal to 0)
Eliminate A

(B) 4
Plug n = 4 into the given information to get: When 10 is divided by 4, the remainder is 4 - 4 (aka 0)
This is not true. When we divide 10 by 4, we get reminder 2
Eliminate B

(C) 7
Plug n = 7 into the given information to get: When 10 is divided by 7, the remainder is 7 - 4 (aka 3)
WORKS!

-------------------------

APPROACH #2: Apply the rule for rebuilding the dividend

There's a nice rule that says, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3

Given: When 10 is divided by the positive integer n, the remainder is n - 4
Since we're not told what the quotient is, let's just say that it's q.
In other words: When 10 is divided by the positive integer n, the quotient is q, and the remainder remainder is n - 4

When we apply the above rule we get: 10 = nq + (n-4)
Add 4 to both sides of the equation to get: 14 = nq + n
Factor to get: 14 = n(q + 1)

Importance: Since n and (q + 1) are INTEGERS, n must be a FACTOR of 14.

Check the answer choices.... only answer choice C (7) is a factor of 14

Cheers,
Brent

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