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Remainder Question

by topspin20 » Sun Sep 01, 2013 10:15 am
When positive integer n is divided by 7, the remainder is 2. When n is divided by 9,the remainder is 3. What is the smallest positive integer k such that k+n is a multiple of 16?

A. 1
B. 2
C. 4
D. 6
E. 9
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by [email protected] » Sun Sep 01, 2013 12:26 pm
Hi topspin20,

In these types of questions, it's really helpful to TEST values.

N/7 = _r2
N could be 2, 9, 16, 23, 30, etc.

N/9 = _r3
N could be 3, 12, 21, 30, etc.

Looking at the two sets of options, the smallest that N could be is 30

So with 30, what is that smallest number that you can add and end up with a multiple of 16?

30 + 2 = 32 = (16 x 2)

Final answer: B

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by GMATGuruNY » Sun Sep 01, 2013 12:33 pm
The problem above is modeled after Q68 in the OG for Quant:

https://www.beatthegmat.com/positive-int ... 84500.html
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by Brent@GMATPrepNow » Sun Sep 01, 2013 12:52 pm
topspin20 wrote:When positive integer n is divided by 7, the remainder is 2. When n is divided by 9,the remainder is 3. What is the smallest positive integer k such that k+n is a multiple of 16?

A. 1
B. 2
C. 4
D. 6
E. 9
The correct answer here is A

Let's first find dome possible values of n. To do so, we'll use a nice rule that says:
If N divided by D, leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

When positive integer n is divided by 7, the remainder is 2.
So, some possible values of n are: 2, 9, 16, 23, 30, 37, 44, etc

When n is divided by 9, the remainder is 3.
So, some possible values of n are: 3, 12, 21, 30, 39, 48, etc

We're looking for an n value that satisfies both conditions. So, since both lists have 30 in them, one possible value of n is 30.

Aside: If we add 2 to 30, we get 32, which is a multiple of 12. So, k=2 satisfies the given condition. So, at this point, we know that the correct answer is either A or B

IMPORTANT: The two lists of possible n-values the result of dividing by 7 and by 9.
Since the least common multiple (LCM) of 7 and 9 is 63, other possible values of n (that satisfy BOTH conditions) can be found by taking 30, and adding multiples of 63.

So, for example, another possible value of n is 93, (30 + 63 = 93)
To make 93 a multiple of 16, we need to add 3 (since 96 is a multiple of 16)

Another possible value of n is 156, (30 + (2)(63) = 156)
To make 156 a multiple of 16, we need to add 4 (since 160 is a multiple of 16)

As you might guess, each subsequent value of n will require us to add an even larger value of k in order to get a multiple of 16. HOWEVER, at some point, we will reach a value of n that is a multiple of 16. In fact, that number is 912 (30 + (14)(63) = 912)

So, if we move up one more number, we get 975 (30 + (15)(63) = 975)
So, 975 is a possible value of n, and if we add 1, (i.e., k = 1), we get 976, which is a multiple of 16.

Answer: A

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by sanjoy18 » Sun Sep 01, 2013 1:15 pm
n=7q+2..(A)
n=9m+3..(B)

thats mean n is the number that satisfy both (A) and (B)

therefore n would be in the form below
n=30+ 63p where p=0,1,2,3....
so smallest value would be when p=0
i.e n=30

hence smallest k should be 2
hence b
Last edited by sanjoy18 on Sun Sep 01, 2013 9:42 pm, edited 1 time in total.
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by vinay1983 » Sun Sep 01, 2013 9:12 pm
I did this as below:

n when divided by 7 leaves remainder 2

So n= 9, 16, 23, 30, 37 etc

n when divided by 9 leaves remainder 3

So n= 12, 21, 30, 39, 48 etc

Here N = 30 is commom for both
Let us assumen n=30

Is n+k mu;tiple of 16
i.e is n+k = 16, 32, 48, 64 etc

If n=30, then the smallest positive integer "K" such that "k+n" is multiple of 16, should be:

We will start with 0

30+0=30 not a multiple of 16

30+1=31 not a multiple of 16

30+2=32 yes a multiple of 16

Let us test answer choices

30+4=34 not a multiple of 16
30+6=36 not a multiple of 16
30+9=39 not a multiple of 16

hence only 2 satisfies the requirement of the smallest positive integer k

B is the option.
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by faraz_jeddah » Mon Sep 02, 2013 6:50 am
Brent@GMATPrepNow wrote:
topspin20 wrote:When positive integer n is divided by 7, the remainder is 2. When n is divided by 9,the remainder is 3. What is the smallest positive integer k such that k+n is a multiple of 16?

A. 1
B. 2
C. 4
D. 6
E. 9
The correct answer here is A

Let's first find dome possible values of n. To do so, we'll use a nice rule that says:
If N divided by D, leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

When positive integer n is divided by 7, the remainder is 2.
So, some possible values of n are: 2, 9, 16, 23, 30, 37, 44, etc

When n is divided by 9, the remainder is 3.
So, some possible values of n are: 3, 12, 21, 30, 39, 48, etc

We're looking for an n value that satisfies both conditions. So, since both lists have 30 in them, one possible value of n is 30.

Aside: If we add 2 to 30, we get 32, which is a multiple of 12. So, k=2 satisfies the given condition. So, at this point, we know that the correct answer is either A or B

IMPORTANT: The two lists of possible n-values the result of dividing by 7 and by 9.
Since the least common multiple (LCM) of 7 and 9 is 63, other possible values of n (that satisfy BOTH conditions) can be found by taking 30, and adding multiples of 63.

So, for example, another possible value of n is 93, (30 + 63 = 93)
To make 93 a multiple of 16, we need to add 3 (since 96 is a multiple of 16)

Another possible value of n is 156, (30 + (2)(63) = 156)
To make 156 a multiple of 16, we need to add 4 (since 160 is a multiple of 16)

As you might guess, each subsequent value of n will require us to add an even larger value of k in order to get a multiple of 16. HOWEVER, at some point, we will reach a value of n that is a multiple of 16. In fact, that number is 912 (30 + (14)(63) = 912)

So, if we move up one more number, we get 975 (30 + (15)(63) = 975)
So, 975 is a possible value of n, and if we add 1, (i.e., k = 1), we get 976, which is a multiple of 16.

Answer: A

Cheers,
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Surely the GMAT is not going to ask us to compute till 975... is it? :shock:
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by Brent@GMATPrepNow » Mon Sep 02, 2013 6:55 am
faraz_jeddah wrote:
Surely the GMAT is not going to ask us to compute till 975... is it? :shock:
We really don't need to keep calculating all the way to 975.

Once we recognize that, with each multiple of 63 we add to n, the value of k increases by 1, we should be able to predict that we'll go from k = 2, to k = 3, to k = 4, . . . k = 15, to k = 0, to k = 1

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by faraz_jeddah » Mon Sep 02, 2013 10:49 am
Thanks Brent for that insight.

Original poster - What is the OA?
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by Brent@GMATPrepNow » Mon Sep 02, 2013 1:52 pm
The OA is definitely A.
First, 975 satisfies the two conditions.
975 divided by 7 = 139 with remainder 2
975 divided by 9 = 108 with remainder 3
So, n could equal 975

The question ask, "What is the smallest positive integer k such that k+n is a multiple of 16?"
If we add 1 to 975, we get 976
976 divided by 16 = 61 with no remainder. In other words, (1 + 975) is a multiple of 16

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by vinay1983 » Mon Sep 02, 2013 10:56 pm
Same question 3 different answers. Who is correct? I request the original poster "topspin20" to reveal the OA?
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by kevincanspain » Tue Sep 03, 2013 1:54 am
As Mitch said, this is a copy of an official question, and a bad copy at that. Different but valid values of n would yield different answers to the question. For A to be the correct answer, the question would have to rephrased, perhaps as follows:

When positive integer n is divided by 7, the remainder is 2. When n is divided by 9,the remainder is 3. What is the smallest positive integer k such that k+n could be a multiple of 16?
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by ganeshrkamath » Tue Sep 03, 2013 3:01 am
topspin20 wrote:When positive integer n is divided by 7, the remainder is 2. When n is divided by 9,the remainder is 3. What is the smallest positive integer k such that k+n is a multiple of 16?

A. 1
B. 2
C. 4
D. 6
E. 9
n = 7a + 2 = 9b + 3
a = (9b+1)/7
b = 3, a = 4
b = 10, a = 13
..

n = 30, 93, 156, 219,...
k = 02, 03, 04, 05, ..., 15, 00, 01, ...

Smallest positive integer k which satisfies the given condition = 1

Choose A

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by vinay1983 » Tue Sep 03, 2013 3:28 am
kevincanspain wrote:As Mitch said, this is a copy of an official question, and a bad copy at that. Different but valid values of n would yield different answers to the question. For A to be the correct answer, the question would have to rephrased, perhaps as follows:

When positive integer n is divided by 7, the remainder is 2. When n is divided by 9,the remainder is 3. What is the smallest positive integer k such that k+n could be a multiple of 16?
Thanks kevin,

I cross checked both the versions, and indeed there is a wording issue in the question.

When positive integer n divide by 5 the remainder is 1. when n is divided by 7, the remainder is 3. What is the smallest integer K such that K+n is a multiple of 35.

a) 3
b) 4
c) 12
d) 32
e) 35 Here the answer was 4 i.e B OA is also B

This(the question in discussion) question's wording is similar to the one above.

This is really confusing!
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