Remainder Question

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topspin20: Junior | Next Rank: 30 Posts; Posts: 25; Joined: Mon Jul 29, 2013 10:17 am

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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Source: Beat The GMAT — Problem Solving | Sun Sep 01, 2013 10:15 am

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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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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

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

Last edited by GMATGuruNY on Tue Sep 03, 2013 3:24 am, edited 1 time in total.

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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

Cheers,
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sanjoy18: Senior | Next Rank: 100 Posts; Posts: 81; Joined: Tue Jun 11, 2013 10:24 pmFollowed by: 1 member; Thanked: 7 times

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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vinay1983: Legendary Member; Posts: 643; Joined: Wed Aug 14, 2013 4:27 amFollowed by: 7 members; Thanked: 48 times

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.

You can, for example never foretell what any one man will do, but you can say with precision what an average number will be up to!

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faraz_jeddah: Master | Next Rank: 500 Posts; Posts: 358; Joined: Thu Apr 18, 2013 9:46 amFollowed by: 7 members; Location: Jeddah, Saudi Arabia; Thanked: 42 times; GMAT Score:730

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,
Brent

Surely the GMAT is not going to ask us to compute till 975... is it?

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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?

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

Cheers,
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Brent Hanneson - Creator of GMATPrepNow.com

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vinay1983: Legendary Member; Posts: 643; Joined: Wed Aug 14, 2013 4:27 amFollowed by: 7 members; Thanked: 48 times

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?

You can, for example never foretell what any one man will do, but you can say with precision what an average number will be up to!

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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?

Kevin Armstrong
GMAT Instructor
Gmatclasses
Madrid

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ganeshrkamath: Master | Next Rank: 500 Posts; Posts: 283; Joined: Sun Jun 23, 2013 11:56 pmFollowed by: 26 members; Location: Bangalore, India; Thanked: 97 times; GMAT Score:750

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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vinay1983: Legendary Member; Posts: 643; Joined: Wed Aug 14, 2013 4:27 amFollowed by: 7 members; Thanked: 48 times

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!

You can, for example never foretell what any one man will do, but you can say with precision what an average number will be up to!

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