Find all School-related info fast with the new School-Specific MBA Forum

It is currently 21 Aug 2014, 12:39

Close

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
Your Progress

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.

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

Find the remainder when 12^190 is divided by 1729 ?

  Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:
Manager
Manager
avatar
Joined: 14 Dec 2011
Posts: 66
Followers: 0

Kudos [?]: 10 [0], given: 66

GMAT Tests User
Find the remainder when 12^190 is divided by 1729 ? [#permalink] New post 23 Sep 2013, 10:37
00:00
A
B
C
D
E

Difficulty:

  35% (medium)

Question Stats:

53% (01:46) correct 47% (00:47) wrong based on 17 sessions
Find the remainder when 12^190 is divided by 1729 ?

A. 12
B. 1
C. 1728
D. 1717
E. 4

Hi guys,

Following is a remainder question (conceptual). Please help me in understanding its concept.

Looking forward to your replies.

Regards
Vinni
[Reveal] Spoiler: OA

Last edited by Bunuel on 25 Sep 2013, 00:43, edited 3 times in total.
Topic Moved. Always post the topic in relevant forum
Expert Post
2 KUDOS received
Magoosh GMAT Instructor
User avatar
Joined: 28 Dec 2011
Posts: 2026
Followers: 486

Kudos [?]: 1986 [2] , given: 30

Re: Find the remainder [#permalink] New post 23 Sep 2013, 13:14
2
This post received
KUDOS
Expert's post
vinnik wrote:
Hi guys,

Following is a remainder question (conceptual). Please help me in understanding its concept.

Find the remainder when 12^190 is divided by 1729 ?

A. 12
B. 1
C. 1728
D. 1717
E. 4

Answer is 1717

Looking forward to your replies.

Regards
Vinni

Dear Vinni,
I'm happy to respond. :-) The first thing I'll say --- this is a couple notches harder than what the GMAT will expect you to know about remainders. For example, here are a couple blogs that covers what the GMAT does expect you to know:
http://magoosh.com/gmat/2012/gmat-quant ... emainders/
http://magoosh.com/gmat/2013/gmat-quant ... questions/

Also, as it turns out, this divisor, 1729, is a number with a famous history in mathematics:
https://en.wikipedia.org/wiki/1729_(number)

Here's how I would approach it.

Notice that 12^3 = 1728, so this divisor is 1729 = ((12^3) + 1). We will use that to our advantage.

12^190 = (12^3)*(12^187) = (12^3)*(12^187) + (12^187) - (12^187)
12^190 = [(12^3)+1]*(12^187) - (12^187)
12^190 = [(12^3)+1]*(12^187) - (12^3)*(12^184)
12^190 = [(12^3)+1]*(12^187) - (12^3)*(12^184) - (12^184) + (12^184)
12^190 = [(12^3)+1]*(12^187) - [(12^3)+1]*(12^184) + (12^184)
12^190 = (1729)*(12^187) - (1729)*(12^184) + (12^184)
The two purple terms are divisible by 1729, so when divided by 1729, they will have a remainder of zero. The green term, when divided by 1729, will have the same remainder as does 12^190 when divided by 1729. That's interesting --- we can use this trick to create a smaller number with the same remainder.

Notice, we could repeat this trick, and bring the number down by a factor of 12^6 again and again. The number 180 is certainly divisible by 6, so 186 must be----- we could drop the power of 12 from 12^190 all the way down to 12^4, that is, 186 powers lower, and we would still have the same remainder when divided by 1729. So, now the whole problem reduces to --- what is the remainder when 12^4 is divided by 1729?

12^4 = (12^3)*(12) = (12^3)*(12) + 12 - 12 = [(12^3)+1]*(12) - 12

So, when 12^4 is divided by 1729, we get the same remainder as when -12 is divided by 1729. OK, that's a little confusing, to have a negative dividend, but when we have one number with a certain remainder, all we have to do is add or subtract the divisor (or a multiple of the divisor) to get other numbers with t the same remainder. Here, I will just add 1729

(-12) + 1729 = 1717

Of course, 1717 < 1729, so when 1717 is divided by 1729, 1729 goes into it zero times with a remainder of 1717. That's the answer.

Does all this make sense?

Mike :-)
_________________

Mike McGarry
Magoosh Test Prep

Image

Image

Manager
Manager
avatar
Joined: 14 Dec 2011
Posts: 66
Followers: 0

Kudos [?]: 10 [0], given: 66

GMAT Tests User
Re: Find the remainder [#permalink] New post 24 Sep 2013, 20:50
Mike,

Thanks for your reply and for posting the following links:-

http://magoosh.com/gmat/2013/gmat-quant ... questions/

http://magoosh.com/gmat/2012/gmat-quant ... emainders/

The first link is really harder (800+ practice questions), but if anyone understands these kinds of questions, then it becomes a lot easier to solve anything of this type.

Above all, one must be familiar with the concept of cyclicity to solve these.

Appreciate it.

Regards
Vinni
Intern
Intern
avatar
Status: Preparation
Joined: 03 Apr 2012
Posts: 8
Location: India
GMAT 1: 700 Q50 V34
GPA: 2.9
Followers: 0

Kudos [?]: 6 [0], given: 10

Re: Find the remainder when 12^190 is divided by 1729 ? [#permalink] New post 25 Sep 2013, 01:38
vinnik wrote:
Find the remainder when 12^190 is divided by 1729 ?

A. 12
B. 1
C. 1728
D. 1717
E. 4

Hi guys,

Following is a remainder question (conceptual). Please help me in understanding its concept.

Looking forward to your replies.

Regards
Vinni


Hi vinnik,

This problem can be solved using remainder theorm.
http://www.pagalguy.com/news/cat-2012-q ... -a-8795953. Here's the explanation for the remainder theorm

12^(190) can be written as. ((12^3)^63)* 12. 12^3 when divided by 1729 gives a remainder -1. so in the numerator we have -12. Now acccording to remainder theorm the answer will be 1729-12=1717.
1 KUDOS received
Intern
Intern
avatar
Joined: 11 Jan 2012
Posts: 9
Location: India
Concentration: Operations, Finance
GMAT Date: 08-31-2012
GPA: 3.7
Followers: 0

Kudos [?]: 9 [1] , given: 11

GMAT ToolKit User
Re: Find the remainder when 12^190 is divided by 1729 ? [#permalink] New post 11 Oct 2013, 20:38
1
This post received
KUDOS
Yes, this problem can be solved by utilizing the remainder theorem. To apply the remainder theorem, attempt to express the numerator such that it the remainder when divided by the denominator is +1 or -1.

So, in this case, we will express 12^190 as (12^3)^63X12. That is because we know 12^3 = 1728 (which is 1 less than the denominator). So now the expression becomes

((12^3)^63x12)/1728 = (-1)^63x12

-1^ odd number = -1. Therefore the expression becomes = -12.

Hence remainder = 1729-12 = 1717.
1 KUDOS received
Manager
Manager
avatar
Joined: 29 Aug 2013
Posts: 77
Location: United States
Concentration: Finance, International Business
GMAT 1: 590 Q41 V29
GMAT 2: 540 Q44 V20
GPA: 3.5
WE: Programming (Computer Software)
Followers: 0

Kudos [?]: 19 [1] , given: 24

Re: Find the remainder when 12^190 is divided by 1729 ? [#permalink] New post 12 Oct 2013, 02:01
1
This post received
KUDOS
vinnik wrote:
Find the remainder when 12^190 is divided by 1729 ?

A. 12
B. 1
C. 1728
D. 1717
E. 4

Hi guys,

Following is a remainder question (conceptual). Please help me in understanding its concept.

Looking forward to your replies.

Regards
Vinni



2 things to keep in mind while solving these type of questions

1) The remainder of the form \frac{(a*b*c)}{d} is the product of their individual remainders.

i.e. Remainder of \frac{a}{d} * Remainder of \frac{b}{d} * Remainder of \frac{c}{d}

2) Remainder can be expressed in either positive or negative form for eg. Remainder of \frac{1728}{1729} can be 1728 or -1 (i.e. 1728 - 1729)

Now here the Remainder of expression (12^190)/1729 = Remainder of (((12^3)^ 63) * 12)/1729

= Remainder of (12^3)^63/1729 * Remainder of 12/1729

That will be (Remainder of (1728^63)/1729) * (Remainder of 12/1729)

i.e. (-1)^63 * 12 = -12

Again negative remainder here so a positive remainder will be 1729-12 = 1717

Kudo if the post helps!!!
Re: Find the remainder when 12^190 is divided by 1729 ?   [#permalink] 12 Oct 2013, 02:01
    Similar topics Author Replies Last post
Similar
Topics:
5 Experts publish their posts in the topic What is the remainder when a is divided by 4? ksear 7 19 Dec 2010, 14:22
4 Experts publish their posts in the topic Finding the remainder when dividing negative numbers mrblack 12 06 Jan 2010, 11:36
When N is divided by T , the quotient is S and the remainder RyanDe680 3 09 May 2008, 14:02
Remainder when x divided by 6 Nsentra 1 01 Oct 2006, 10:46
When S is divided by 5 remainder is 3, when it is divided by getzgetzu 16 26 Apr 2006, 02:32
Display posts from previous: Sort by

Find the remainder when 12^190 is divided by 1729 ?

  Question banks Downloads My Bookmarks Reviews Important topics  


GMAT Club MBA Forum Home| About| Privacy Policy| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group and phpBB SEO

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.