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

It is currently 01 Sep 2015, 04:23
GMAT Club Tests

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

What is the remainder when 333^222 is divided by 7?

  Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:
Intern
Intern
avatar
Joined: 12 Mar 2013
Posts: 7
Followers: 0

Kudos [?]: 8 [0], given: 1

What is the remainder when 333^222 is divided by 7? [#permalink] New post 21 Jul 2013, 01:16
11
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

40% (02:26) correct 60% (00:58) wrong based on 434 sessions
What is the remainder when 333^222 is divided by 7?

A. 3
B. 2
C. 5
D. 7
E. 1
[Reveal] Spoiler: OA

Last edited by Bunuel on 21 Jul 2013, 02:30, edited 2 times in total.
Renamed the topic, edited the question and the tags.
Expert Post
3 KUDOS received
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 29172
Followers: 4734

Kudos [?]: 49990 [3] , given: 7518

Re: what is the reminder when 333^222 is divided by 7? [#permalink] New post 21 Jul 2013, 02:27
3
This post received
KUDOS
Expert's post
2
This post was
BOOKMARKED
jonyg wrote:
what is the reminder when 333^222 is divided by 7?
a.3
b.2
c.5
d.7
e.1


official answer=>e

source- random internet


What is the remainder when 333^222 is divided by 7?
A. 3
B. 2
C. 5
D. 7
E. 1

\(333^{222}=(329+4)^{222}=(7*47+4)^{222}\). Now if we expand this, all terms but the last one will have 7*47 as a multiple and thus will be divisible by 7. The last term will be \(4^{222}=2^{444}\). So we should find the remainder when \(2^{444}\) is divided by 7.

2^1 divided by 7 yields remainder of 2;
2^2 divided by 7 yields remainder of 4;
2^3 divided by 7 yields remainder of 1;

2^4 divided by 7 yields remainder of 2;
2^5 divided by 7 yields remainder of 4;
2^6 divided by 7 yields remainder of 1;
...

The remainder repeats in blocks of three: {2-4-1}. So, the remainder of \(2^{444}\) divided by 7 would be the same as \(2^3\) divided by 7 (444 is a multiple of 3). \(2^3\) divided by 7 yields remainder of 1.

Answer: E.

Similar question to practice:
what-is-the-remainder-when-43-86-is-divided-by-134778.html
when-51-25-is-divided-by-13-the-remainder-obtained-is-130220.html
what-is-the-remainder-of-126493.html
what-is-the-remainder-when-32-32-32-is-divided-by-100316.html
what-is-the-remainder-when-18-22-10-is-divided-by-99724.html

Theory on remainders problems: remainders-144665.html

DS remainders problems to practice: search.php?search_id=tag&tag_id=198
PS remainders problems to practice: search.php?search_id=tag&tag_id=199

Hope it helps.
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis ; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) ; 12. Tricky questions from previous years.

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
25 extra-hard Quant Tests

GMAT Club Premium Membership - big benefits and savings

Current Student
avatar
Joined: 03 Aug 2012
Posts: 920
Concentration: General Management, General Management
GMAT 1: 630 Q47 V29
GMAT 2: 680 Q50 V32
GPA: 3.7
WE: Information Technology (Investment Banking)
Followers: 17

Kudos [?]: 386 [0], given: 322

Premium Member
Re: What is the remainder when 333^222 is divided by 7? [#permalink] New post 28 Jul 2013, 03:05
(333)^222

Rem(333/7) = 4

=> (4) ^222

=> (16) ^111

Rem(16/7) = 2

(2)^111 = 2^100 * 2^11

Now let us observe the Rem(2^10)/7

2^10 = 1024 => Rem(1024/7) = 2

REM(2^100 * 2^11 ) be 7 = REM((2^10)^10 * 2^11) by 7 => REM((2)^10 * 2^10 * 2) by 7

=> Rem( 2* 2 *2 ) by 7

=> 8/7

=> 1

(E)
_________________

Rgds,
TGC!
_____________________________________________________________________
I Assisted You => KUDOS Please
_____________________________________________________________________________

Manager
Manager
avatar
Status: Persevering
Joined: 15 May 2013
Posts: 225
Location: India
Concentration: Technology, Leadership
GMAT Date: 08-02-2013
GPA: 3.7
WE: Consulting (Consulting)
Followers: 1

Kudos [?]: 62 [0], given: 34

GMAT ToolKit User
Re: What is the remainder when 333^222 is divided by 7? [#permalink] New post 31 Jul 2013, 03:27
333^222/7

3^222 *(111)^222 /7 =>(111)^222/7=> in terms of remainder (6)^222/7 or (-1)^222/7 which leaves 1 now the other part (3^2)^111/7 => (2)^111/7 =>(8)^27/7=>1^27 and this part is also one .
_________________

--It's one thing to get defeated, but another to accept it.

Intern
Intern
avatar
Joined: 02 May 2012
Posts: 9
Followers: 0

Kudos [?]: 5 [0], given: 5

Re: what is the reminder when 333^222 is divided by 7? [#permalink] New post 01 Aug 2013, 12:37
Bunuel wrote:
jonyg wrote:
what is the reminder when 333^222 is divided by 7?
a.3
b.2
c.5
d.7
e.1


official answer=>e

source- random internet


What is the remainder when 333^222 is divided by 7?
A. 3
B. 2
C. 5
D. 7
E. 1

\(333^{222}=(329+4)^{222}=(7*47+4)^{222}\). Now if we expand this, all terms but the last one will have 7*47 as a multiple and thus will be divisible by 7. The last term will be \(4^{222}=2^{444}\). So we should find the remainder when \(2^{444}\) is divided by 7.

2^1 divided by 7 yields remainder of 2;
2^2 divided by 7 yields remainder of 4;
2^3 divided by 7 yields remainder of 1;

2^4 divided by 7 yields remainder of 2;
2^5 divided by 7 yields remainder of 4;
2^6 divided by 7 yields remainder of 1;
...

The remainder repeats in blocks of three: {2-4-1}. So, the remainder of \(2^{444}\) divided by 7 would be the same as \(2^3\) divided by 7 (444 is a multiple of 3). \(2^3\) divided by 7 yields remainder of 1.

Answer: E.


Hey can you explain to me how you get a remainder of 2 when you divide 2^1/7?
Expert Post
Verbal Forum Moderator
Verbal Forum Moderator
User avatar
Joined: 10 Oct 2012
Posts: 629
Followers: 64

Kudos [?]: 772 [0], given: 135

Premium Member
Re: what is the reminder when 333^222 is divided by 7? [#permalink] New post 02 Aug 2013, 03:28
Expert's post
iNumbv wrote:
Bunuel wrote:
jonyg wrote:
what is the reminder when 333^222 is divided by 7?
a.3
b.2
c.5
d.7
e.1


official answer=>e

source- random internet

Hey can you explain to me how you get a remainder of 2 when you divide 2^1/7?


This might help : remainders-144665.html

If x and y are positive integers, there exist unique integers q and r, called the quotient and remainder, respectively, such that \(y =divisor*quotient+remainder\)= xq + r and \(0\leq{r}<x.\)

For example, when 15 is divided by 6, the quotient is 2 and the remainder is 3 since 15 = 6*2 + 3.

Notice that \(0\leq{r}<x\) means that remainder is a non-negative integer and always less than divisor.

As for your query, we can write \(2 = 0*7+2\), where 7 is the divisor, and 2 is the remainder.

Hope this helps.
_________________

All that is equal and not-Deep Dive In-equality

Hit and Trial for Integral Solutions

Intern
Intern
avatar
Joined: 02 Jun 2013
Posts: 19
Followers: 0

Kudos [?]: 8 [0], given: 74

Re: What is the remainder when 333^222 is divided by 7? [#permalink] New post 02 Aug 2013, 04:18
Hi,
My solution is as follows when 333/7 gives reminder 4 thus we have to find out 4^222 now 7 is a prime no so according to fermants littile therom (4^6)/7=1 now we have to see if 222 is divisble by 6 thus 222=6*37 hence 4^6k/7 =1 hence answer is 1 ie e
Manager
Manager
avatar
Joined: 14 Nov 2008
Posts: 70
Followers: 2

Kudos [?]: 20 [0], given: 1

Re: What is the remainder when 333^222 is divided by 7? [#permalink] New post 30 Aug 2013, 10:58
A simple one line solution to this problem can be this:
Rem(333^222)/7 = Rem(4^222)/7 = Rem(64^74)/7=Rem((63+1)^74)/7 = 1
Expert Post
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 29172
Followers: 4734

Kudos [?]: 49990 [0], given: 7518

Re: What is the remainder when 333^222 is divided by 7? [#permalink] New post 09 Mar 2014, 12:14
Expert's post
Intern
Intern
avatar
Joined: 02 Jul 2013
Posts: 4
Followers: 0

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

Re: What is the remainder when 333^222 is divided by 7? [#permalink] New post 29 Apr 2014, 13:36
Hi,
I am confused between 2 approaches for these kinds of problems
Approach 1: Binomial Theorem.
Approach 2: Find the unit's digit of the exponent and then find the remainder.

Unit's digit of 333^222 = unit's digit of 3^222. Then divide that by 7.
Cyclicity of 3 = 4 {3,9,7,1}. 222/3 has a remainder of 2. 3^2 has a unit's digit of 9. 9/7 has a remainder of 2

In Approach 2, i don't always get the same ans as by using Approach 1. Which approach is preferred for these kinds of problems?

Thanks.
Senior Manager
Senior Manager
avatar
Joined: 27 Oct 2013
Posts: 261
Location: India
Concentration: General Management, Technology
GMAT Date: 03-02-2015
GPA: 3.88
Followers: 0

Kudos [?]: 96 [0], given: 79

Re: What is the remainder when 333^222 is divided by 7? [#permalink] New post 29 Apr 2014, 19:39
Hi All,

I used the following approach.

(333^222)/7

(333/7) = Remainder is 4

4^222 can be written as 2^444 which can be written as (2^3)^148

now what we have to do find is

((2^3)^148)/7

we can write the above expression as

((7+1)^148)/7

now apply remainder theorem.

Hence Remainder is 1.

Option E is correct
Expert Post
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 29172
Followers: 4734

Kudos [?]: 49990 [0], given: 7518

Re: What is the remainder when 333^222 is divided by 7? [#permalink] New post 30 Apr 2014, 06:44
Expert's post
1
This post was
BOOKMARKED
gmatcracker2407 wrote:
Hi,
I am confused between 2 approaches for these kinds of problems
Approach 1: Binomial Theorem.
Approach 2: Find the unit's digit of the exponent and then find the remainder.

Unit's digit of 333^222 = unit's digit of 3^222. Then divide that by 7.
Cyclicity of 3 = 4 {3,9,7,1}. 222/3 has a remainder of 2. 3^2 has a unit's digit of 9. 9/7 has a remainder of 2

In Approach 2, i don't always get the same ans as by using Approach 1. Which approach is preferred for these kinds of problems?

Thanks.


The units digit does not determine the remainder when dividing by 7. For example, 9 divided by 7 gives the remainder of 2, 19 divided by 7 gives the remainder of 5, 29 divided by 7 gives the remainder of 1, ...
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis ; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) ; 12. Tricky questions from previous years.

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
25 extra-hard Quant Tests

GMAT Club Premium Membership - big benefits and savings

GMAT Club Legend
GMAT Club Legend
User avatar
Joined: 09 Sep 2013
Posts: 6151
Followers: 342

Kudos [?]: 70 [0], given: 0

Premium Member
Re: What is the remainder when 333^222 is divided by 7? [#permalink] New post 15 Jul 2015, 21:01
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

GMAT Books | GMAT Club Tests | Best Prices on GMAT Courses | GMAT Mobile App | Math Resources | Verbal Resources

Manager
Manager
avatar
Joined: 06 Mar 2014
Posts: 89
Followers: 0

Kudos [?]: 4 [0], given: 2

Re: What is the remainder when 333^222 is divided by 7? [#permalink] New post 25 Jul 2015, 04:43
Hi VeritasPrepKarishma :
Bunuel
Can you please solve my doubt,
From 4^222 , 222 Is basically 55m+2. Since 4 has a cyclicity of { 4, 6} , the unit's digit here will be 6.
When you divide this by 7, the remainder will be 6.
But answer says remainder will be 1.

Can you please help.
Verbal Forum Moderator
Verbal Forum Moderator
avatar
Joined: 02 Aug 2009
Posts: 1230
Followers: 27

Kudos [?]: 511 [0], given: 19

Premium Member CAT Tests
Re: What is the remainder when 333^222 is divided by 7? [#permalink] New post 25 Jul 2015, 05:01
Shree9975 wrote:
Hi VeritasPrepKarishma :
Bunuel
Can you please solve my doubt,
From 4^222 , 222 Is basically 55m+2. Since 4 has a cyclicity of { 4, 6} , the unit's digit here will be 6.
When you divide this by 7, the remainder will be 6.
But answer says remainder will be 1.

Can you please help.


Hi,
the units digit cannot determine the remainder except in the case of 2,5,10 etc...
6 will have remainder 6 but 16 will have 2 and so on..
the right way would be 4^222=(4^3)^74...
now 4^3=64 and the remainder will be 1 when divided by 7..
so ans will be1^74=1
1 is the remainder..
hope it helps
Re: What is the remainder when 333^222 is divided by 7?   [#permalink] 25 Jul 2015, 05:01
    Similar topics Author Replies Last post
Similar
Topics:
10 Experts publish their posts in the topic What is the remainder when you divide 2^200 by 7? g3kr 14 16 Oct 2012, 19:42
4 Experts publish their posts in the topic What is the remainder when you divide 2^200 by 7? carcass 7 02 Oct 2011, 10:50
56 Experts publish their posts in the topic What is the remainder when 32^32^32 is divided by 7? gurpreetsingh 56 02 Sep 2010, 17:23
23 Experts publish their posts in the topic What is the remainder when (18^22)^10 is divided by 7 ? Financier 19 24 Aug 2010, 01:35
1 What is the remainder when 7^74 - 5^74 is divided by 24? wizardofwashington 10 03 Jul 2008, 12:19
Display posts from previous: Sort by

What is the remainder when 333^222 is divided by 7?

  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®.