GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 22 Feb 2020, 20:46 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # A positive integer is divisible by 9 if and only if the sum of its

Author Message
TAGS:

### Hide Tags

Intern  Joined: 10 Feb 2016
Posts: 6
Location: Finland
Concentration: General Management, Strategy
WE: Information Technology (Computer Software)
A positive integer is divisible by 9 if and only if the sum of its  [#permalink]

### Show Tags

12
1
78 00:00

Difficulty:   25% (medium)

Question Stats: 81% (02:17) correct 19% (02:12) wrong based on 1324 sessions

### HideShow timer Statistics

A positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. If n is a positive integer, for which of the following values of k is $$25*10^n + k*10^{2n}$$ divisible by 9?

(A) 9
(B) 16
(C) 23
(D) 35
(E) 47

Originally posted by bigdady on 14 Mar 2016, 12:34.
Last edited by Bunuel on 17 Jun 2018, 11:59, edited 3 times in total.
Formatted the question
Target Test Prep Representative V
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 9443
Location: United States (CA)
Re: A positive integer is divisible by 9 if and only if the sum of its  [#permalink]

### Show Tags

13
4
A positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. If n is a positive integer, for which of the following values of k is 25*10^n + k*10^2n divisible by 9?

(A) 9
(B) 16
(C) 23
(D) 35
(E) 47

We need to determine for which value of k 25*10^n + k*10^2n is divisible by 9.

We see that 10^n and 10^2n will always have a digit of 1 and then zeros. So, excluding k, the sum of the digits in our expression is 2 + 5 = 7 (since (25)(10^n) has a 2, 5, and zeros).

We need to determine, of our answer choices, which when added to 7 will produce a sum that is divisible by 9. Scanning our answer choices, we see that 47 is the correct answer.

2 + 5 + 4 + 7 = 18, which is divisible by 9.

_________________

# Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Intern  Joined: 14 Mar 2016
Posts: 3
Location: United States
Concentration: Technology, Strategy
Schools: Sloan '19
GPA: 3.2
Re: A positive integer is divisible by 9 if and only if the sum of its  [#permalink]

### Show Tags

10
3
For any value of N, when you multiply by 10ˆn or 10ˆ2n you will be only adding zeros. What you only need to do is check if the sum of 2+5+the other possible values for k in the question add up to 9. The only possible answer is E.

Originally posted by marcelonac on 14 Mar 2016, 14:06.
Last edited by marcelonac on 15 Mar 2016, 12:11, edited 1 time in total.
##### General Discussion
Intern  B
Joined: 16 Apr 2015
Posts: 30
Re: A positive integer is divisible by 9 if and only if the sum of its  [#permalink]

### Show Tags

1
1
marcelonac wrote:
For any value of N, the sum of the digits of 10ˆn and 10ˆ2n will always be 1. Since they're being multiplied by 25 and k, what you only need to do is check if the sum of 2+5+the other possible valued for k in the question add up to 9. The only possible answer is E.

really neat explainantion +1Kudo
Current Student D
Joined: 12 Aug 2015
Posts: 2535
Schools: Boston U '20 (M)
GRE 1: Q169 V154 Re: A positive integer is divisible by 9 if and only if the sum of its  [#permalink]

### Show Tags

5
Excellent Question
Here Taking out 10^n common => 100k+25 must be divisible by 9
checking values (try and start with last in such questions of plugging in values ) we get => E
_________________
Intern  B
Joined: 19 Sep 2012
Posts: 10
Re: A positive integer is divisible by 9 if and only if the sum of its  [#permalink]

### Show Tags

3
Here is my approach:

It tells us that 25 × 10^n + k × 10^(2n) is divisible by 9 and we know that 10^whatever is not divisible by 9. So we just plug in numbers to find a number that satisfies that the sum of its digits (25 + k) is divisible by 9.
Starting with option E: 25 + 47 = 72 and 7 + 2 = 9. Hence, when k=47 the number is divisible by 9.
Senior Manager  B
Joined: 13 Oct 2016
Posts: 352
GPA: 3.98
Re: A positive integer is divisible by 9 if and only if the sum of its  [#permalink]

### Show Tags

2
A positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. If n is a positive integer, for which of the following values of k is 25*10^n + k*10^2n divisible by 9?

(A) 9
(B) 16
(C) 23
(D) 35
(E) 47

25 = 7 (mod 9) = -2 (mod 9)

$$10^n = 10^{2n}$$ = 1 (mod 9)

-2 * 1 + k*1 = k - 2

k should have remainder 2 when divided by 9 to give us total remainder 0.

Only option which leaves remainder 2 upon division by 9 is 47 (E).
Manager  B
Joined: 09 Aug 2016
Posts: 61
Re: A positive integer is divisible by 9 if and only if the sum of its  [#permalink]

### Show Tags

3
This looks like >620 imho

Solution:

25*10^n + k*10^2n = (25 * 10^n) + (k* 10^n * 10^n) {you should be able to see that 10^2n breaks into 10^n * 10^n else work your exponents roots in algebra}

then you can do 10^n * [ 25 +(k * 10^n) ] And here the logic/critical thinking begins

a) The only thing that the very first 10^n does to the number within the square brackets is simply padding it with zero at the end. (as other users suggested before). So it doesn't play any role to the divisibility hence can be fully ignored (N.B. n>0).

b) now 25 + (k * 10^n) simply is K * mul(10) + 25 therefore you care only about the sum of the digits K, 2 and 5. By using "back-solving" you can find E

Therefore E.

good luck!
Intern  B
Joined: 07 Mar 2016
Posts: 25
Location: United States
GMAT 1: 710 Q50 V35
GPA: 3.2
Re: A positive integer is divisible by 9 if and only if the sum of its  [#permalink]

### Show Tags

stonecold wrote:
Excellent Question
Here Taking out 10^n common => 100k+25 must be divisible by 9
checking values (try and start with last in such questions of plugging in values ) we get => E

nice explanation stonecold, thank you Manager  B
Joined: 09 Aug 2016
Posts: 61
Re: A positive integer is divisible by 9 if and only if the sum of its  [#permalink]

### Show Tags

Korhand wrote:
stonecold wrote:
Excellent Question
Here Taking out 10^n common => 100k+25 must be divisible by 9
checking values (try and start with last in such questions of plugging in values ) we get => E

nice explanation stonecold, thank you I think the above way misses a crucial point.

Since n>0 then 10^n can be 10 and not 100 (i.e. n = 1 since n>0). Therefore the original relationship deduces to k * 10. So in order to be on the safe side I recommend to solve fro two values (more than two is an overkill imho)

10* k
100 * k
Manager  B
Joined: 26 Mar 2017
Posts: 105
Re: A positive integer is divisible by 9 if and only if the sum of its  [#permalink]

### Show Tags

2
I'm just using ridiculous numbers for these questions

so let n be= 10000000000000000

then we get something like:

250000000000.....+k00000000000000000000........

from this it becomes clear that we just have to look for 2+5 + k divisible by 9

7+47=54 which is divisible by 9
_________________
I hate long and complicated explanations!
Intern  B
Status: London UK GMAT Consultant / Tutor
Joined: 30 Oct 2012
Posts: 49
Re: A positive integer is divisible by 9 if and only if the sum of its  [#permalink]

### Show Tags

3
2
Hi GMATters,

Here's my video explanation of this question:

Enjoy!

Rowan
_________________
Is Your GMAT Score Stuck in the 600s? This FREE 8-Video, 20-Page Guide Can Help.

http://yourgmatcoach.com/gmat-score-stuck-plateau-600/

PS have you seen the new GMAT Work and Rates guide? Comes with a free 8-video course.

https://yourgmatcoach.podia.com/courses/how-to-beat-gmat-work-and-rates-problems

Originally posted by ygcrowanhand on 23 May 2017, 07:28.
Last edited by Bunuel on 28 Sep 2017, 07:39, edited 3 times in total.
Edited. S
Status: It's now or never
Joined: 10 Feb 2017
Posts: 172
Location: India
GMAT 1: 650 Q40 V39
GPA: 3
WE: Consulting (Consulting)
Re: A positive integer is divisible by 9 if and only if the sum of its  [#permalink]

### Show Tags

1
Thanks for the explanation, very helpful. The youtube link doesn't work, just wanted to let you know ygcrowanhand
_________________

Class of 2019: Mannheim Business School
Class 0f 2020: HHL Leipzig
Math Expert V
Joined: 02 Sep 2009
Posts: 61396
Re: A positive integer is divisible by 9 if and only if the sum of its  [#permalink]

### Show Tags

AnubhavK wrote:
Thanks for the explanation, very helpful. The youtube link doesn't work, just wanted to let you know ygcrowanhand

_________________
IIMA, IIMC School Moderator V
Joined: 04 Sep 2016
Posts: 1387
Location: India
WE: Engineering (Other)
Re: A positive integer is divisible by 9 if and only if the sum of its  [#permalink]

### Show Tags

ScottTargetTestPrep Bunuel mikemcgarry IanStewart shashankism Engr2012

Quote:
We see that 10^n and 10^2n will always have a digit of 1 and then zeros. So, excluding k, the sum of the digits in our expression is 2 + 5 = 7 (since (25)(10^n) has a 2, 5, and zeros).

Why do we not add for 10^x here and only consider 2 + 5 = 7
_________________
It's the journey that brings us happiness not the destination.

Feeling stressed, you are not alone!!
CEO  G
Joined: 20 Mar 2014
Posts: 2536
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44 GPA: 3.7
WE: Engineering (Aerospace and Defense)
Re: A positive integer is divisible by 9 if and only if the sum of its  [#permalink]

### Show Tags

2
ScottTargetTestPrep Bunuel mikemcgarry IanStewart shashankism Engr2012

Quote:
We see that 10^n and 10^2n will always have a digit of 1 and then zeros. So, excluding k, the sum of the digits in our expression is 2 + 5 = 7 (since (25)(10^n) has a 2, 5, and zeros).

Why do we not add for 10^x here and only consider 2 + 5 = 7

Try to see it this way: 25*10 = 250 (sum of digits = 2+5); similarly 25*10^6 = 25000000 (sum of the digits = 2+5) etc. So in all cases, the sum of the digits for 25*(10^n) will always be 2+5 = 7.

Hope this helps.

P.S. the easiest/most straightforward way for this would have been to assume n=1 and then play with smaller numbers.
Intern  B
Joined: 03 Aug 2017
Posts: 6
Re: A positive integer is divisible by 9 if and only if the sum of its  [#permalink]

### Show Tags

What's the level of this question?

Sent from my Redmi Note 4 using GMAT Club Forum mobile app
Math Expert V
Joined: 02 Sep 2009
Posts: 61396
Re: A positive integer is divisible by 9 if and only if the sum of its  [#permalink]

### Show Tags

akshay94raja wrote:
What's the level of this question?

Sent from my Redmi Note 4 using GMAT Club Forum mobile app

The difficulty level of a question is given in the first post.
Attachment: 2017-09-11_1947.png [ 90.83 KiB | Viewed 20329 times ]

_________________
Director  V
Joined: 24 Oct 2016
Posts: 585
GMAT 1: 670 Q46 V36 GMAT 2: 690 Q47 V38 Re: A positive integer is divisible by 9 if and only if the sum of its  [#permalink]

### Show Tags

1
A positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. If n is a positive integer, for which of the following values of k is $$25*10^{n + k}*10^{2n}$$ divisible by 9?

(A) 9
(B) 16
(C) 23
(D) 35
(E) 47

FYI Bunuel bigdady This question has a typo. It should be $$25*10^{n} + k*10^{2n}$$ and not $$25*10^{n + k}*10^{2n}$$.
Math Expert V
Joined: 02 Sep 2009
Posts: 61396
Re: A positive integer is divisible by 9 if and only if the sum of its  [#permalink]

### Show Tags

dabaobao wrote:
A positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. If n is a positive integer, for which of the following values of k is $$25*10^{n + k}*10^{2n}$$ divisible by 9?

(A) 9
(B) 16
(C) 23
(D) 35
(E) 47

FYI Bunuel bigdady This question has a typo. It should be $$25*10^{n} + k*10^{2n}$$ and not $$25*10^{n + k}*10^{2n}$$.

_______________
Edited. Thank you.
_________________ Re: A positive integer is divisible by 9 if and only if the sum of its   [#permalink] 17 Jun 2018, 11:59

Go to page    1   2    Next  [ 25 posts ]

Display posts from previous: Sort by

# A positive integer is divisible by 9 if and only if the sum of its  