Author 
Message 
TAGS:

Hide Tags

Intern
Joined: 05 Mar 2013
Posts: 44
Location: India
Concentration: Entrepreneurship, Marketing
GMAT Date: 06052013
GPA: 3.2

What is the remainder when the number 3^1989 is divided by 7 [#permalink]
Show Tags
17 May 2013, 06:43
3
This post received KUDOS
33
This post was BOOKMARKED
Question Stats:
48% (01:30) correct 52% (01:24) wrong based on 574 sessions
HideShow timer Statistics
What is the remainder when the number 3^1989 is divided by 7? A. 1 B. 5 C. 6 D. 4 E. 3
Official Answer and Stats are available only to registered users. Register/ Login.
_________________
"Kudos" will help me a lot!!!!!!Please donate some!!!
Completed Official Quant Review OG  Quant
In Progress Official Verbal Review OG 13th ed MGMAT IR AWA Structure
Yet to do 100 700+ SC questions MR Verbal MR Quant
Verbal is a ghost. Cant find head and tail of it.
Last edited by Bunuel on 17 May 2013, 16:19, edited 1 time in total.
Edited the question.



Intern
Joined: 04 Jan 2013
Posts: 13
Location: India
Concentration: Finance
GMAT Date: 08262013
GPA: 2.83
WE: Other (Other)

Re: What is the remainder??? [#permalink]
Show Tags
17 May 2013, 06:48
3
This post received KUDOS
2
This post was BOOKMARKED
3 / 7 rem = 3 3^2 = 9 / 7 rem = 2 3 ^ 3 = 27 / 7 rem = 6 or 1 (1)
Now, 1989/3 = 663
From (1) above, 3 ^ 1989 = (3^3) ^ 663 ; rem = (1) ^ 663 = 1 or 6 Ans: 6
Hope it is clear.



Intern
Joined: 05 Mar 2013
Posts: 44
Location: India
Concentration: Entrepreneurship, Marketing
GMAT Date: 06052013
GPA: 3.2

Re: What is the remainder??? [#permalink]
Show Tags
17 May 2013, 06:50
mkdureja wrote: 3 / 7 rem = 3 3^2 = 9 / 7 rem = 2 3 ^ 3 = 27 / 7 rem = 6 or 1 (1) Now, 1989/3 = 663 From (1) above, 3 ^ 1989 = (3^3) ^ 663 = rem = (1) ^ 663 = 1 or 6 Ans: 6
Hope it is clear. +1 Kudos Nice!!!! :D
_________________
"Kudos" will help me a lot!!!!!!Please donate some!!!
Completed Official Quant Review OG  Quant
In Progress Official Verbal Review OG 13th ed MGMAT IR AWA Structure
Yet to do 100 700+ SC questions MR Verbal MR Quant
Verbal is a ghost. Cant find head and tail of it.



Manager
Status: *Lost and found*
Joined: 25 Feb 2013
Posts: 123
Location: India
Concentration: General Management, Technology
GPA: 3.5
WE: Web Development (Computer Software)

Re: What is the remainder??? [#permalink]
Show Tags
17 May 2013, 06:51
SrinathVangala wrote: What is the remainder when the number 3^1989 is divided by 7?
A. 1 B. 5 C. 6 D. 4 E. 3 Answer would be [C] as mentioned. 3^1989 = 3^(3*663) = 27^663 The remainder left by 27/7 will be the same as the remainder left under 27^663. Hence the remainder is 1 or 6. Hope my answer is accurate! Regards, Arpan
_________________
Feed me some KUDOS! *always hungry*
My Thread : Recommendation Letters



Intern
Joined: 05 Apr 2010
Posts: 14

Re: What is the remainder??? [#permalink]
Show Tags
17 May 2013, 07:09
Can someone please explain me in detail how we arrived at the problem? I solved the Q for unit digit of the expression. Is this approach wrong? How to arrive at the solution?



Intern
Joined: 04 Jan 2013
Posts: 13
Location: India
Concentration: Finance
GMAT Date: 08262013
GPA: 2.83
WE: Other (Other)

Re: What is the remainder??? [#permalink]
Show Tags
17 May 2013, 07:19
1
This post received KUDOS
1
This post was BOOKMARKED
coolpintu wrote: Can someone please explain me in detail how we arrived at the problem? I solved the Q for unit digit of the expression. Is this approach wrong? How to arrive at the solution? Unit digit is remainder when divided by 10, what we are asked is remainder when we divide the no. by 7, so finding unit unit digit wont help you. A rule: If a when divided by b leaves remainder c, then, a^x, when divided by b will leave the remainder c^x. So, to approach the problem, we can start from raised to power 1, and go on and stop when we get 1 or 1 as remainder, then it becomes easy to solve it. Like in this case, 3^3 leaves remainder 1 when divided by 7, so using the above rule, we can say that 3^1989 = (3^3)^ 663 will leave remainder (1)^663 or 1, when divided by 7.



Intern
Joined: 06 Jan 2013
Posts: 24
GPA: 3
WE: Engineering (Transportation)

Re: What is the remainder??? [#permalink]
Show Tags
17 May 2013, 09:09
1
This post received KUDOS
1
This post was BOOKMARKED
This is how I solved it: \(\frac{3^{1989}}{7}=\frac{(74)^{1989}}{7}\) Every term in the expansion of \((74)^{1989}\) would contain the number '7' except \((4)^{1989}\) So it ultimately reduces to finding the the remainder when \((4)^{1989}\) is divided by 7. \(\frac{(4)^{1989}}{7}=\frac{(1).(4)^{1989}}{7}=\frac{(1).(64)^{663}}{7}\) Now 64 would leave a remainder of 1 when divided by 7. Hence the final remainder would be = 1x1=1. This is a negative remainder,hence for finding the actual remainder we just have to add this negative remainder to the divisor i.e. 7 Therefore, the final remainder is (1+7)=6
_________________
If you shut your door to all errors, truth will be shut out.



Manager
Joined: 21 Aug 2013
Posts: 99

Re: What is the remainder when the number 3^1989 is divided by 7 [#permalink]
Show Tags
16 Apr 2014, 04:07
Hi Bunuel, Can you explain this.
_________________
Veritas Prep  650 MGMAT 1 590 MGMAT 2 640 (V48/Q31)
Please help the community by giving Kudos.



Math Expert
Joined: 02 Sep 2009
Posts: 43850

Re: What is the remainder when the number 3^1989 is divided by 7 [#permalink]
Show Tags
16 Apr 2014, 04:51
5
This post received KUDOS
Expert's post
10
This post was BOOKMARKED
seabhi wrote: Hi Bunuel, Can you explain this. What is the remainder when the number 3^1989 is divided by 7?A. 1 B. 5 C. 6 D. 4 E. 3 \(3^{1989}=3^{3*663}=27^{663}=(21+6)^{663}\). Now if we expand this, all terms but the last one will have 21 as a multiple and thus will be divisible by 7. The last term will be \(6^{663}\). So we should find the remainder when \(6^{663}\) is divided by 7. 6^1 divided by 7 yields remainder of 6; 6^2 divided by 7 yields remainder of 1; 6^3 divided by 7 yields remainder of 6 again; ... The remainder repeats in blocks of two: {61}{61}{61}... When the power is odd the remainder is 6 and when the power is even the remainder is 1. So, the remainder when \(6^{663}=6^{odd}\) is divided by 7 is 6. Answer: C. Units digits, exponents, remainders problems: newunitsdigitsexponentsremaindersproblems168569.htmlHope it helps.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  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? Extrahard Quant Tests with Brilliant Analytics



NonHuman User
Joined: 09 Sep 2013
Posts: 13810

Re: What is the remainder when the number 3^1989 is divided by 7 [#permalink]
Show Tags
01 May 2015, 18:00
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
Joined: 22 Aug 2014
Posts: 189

Re: What is the remainder when the number 3^1989 is divided by 7 [#permalink]
Show Tags
05 May 2015, 00:13
SrinathVangala wrote: What is the remainder when the number 3^1989 is divided by 7?
A. 1 B. 5 C. 6 D. 4 E. 3 Easy to solve like this: 3^3(663) 27^663 (21+6)^663 21/7 no remainder 6/76 remainder



Intern
Joined: 30 Jun 2015
Posts: 6

Re: What is the remainder when the number 3^1989 is divided by 7 [#permalink]
Show Tags
24 Sep 2015, 12:15
3/7 remainder 3, 9/7 remainder 2, 27/7 remainder 6
so the cyclicity is 3264000 1989/7 gives remainder 1, so remainder for 3^1989 should be 3, what am i missing? Please explain?



Current Student
Joined: 20 Mar 2014
Posts: 2683
Concentration: Finance, Strategy
GPA: 3.7
WE: Engineering (Aerospace and Defense)

Re: What is the remainder when the number 3^1989 is divided by 7 [#permalink]
Show Tags
24 Sep 2015, 16:16
SahilKataria wrote: 3/7 remainder 3, 9/7 remainder 2, 27/7 remainder 6
so the cyclicity is 3264000 1989/7 gives remainder 1, so remainder for 3^1989 should be 3, what am i missing? Please explain? My question to you is: how are you getting 'cyclicity" as 3264000? Cyclicity is defined as number of terms after which a particular pattern will repeat itself be it in remainders or unit's digits etc. How is the cyclicity 32640000 and then based on 1989/7, how can you relate the remainder to what the is asking? For this question, the best approach is Bunuel's at whatistheremainderwhenthenumber31989isdividedby152951.html#p1356693One way to solve these questions is to make sure to express the given exponent in some 'relatable' form wrt the denominator which is what is done above.



Rotman Thread Master
Joined: 15 Mar 2016
Posts: 97
Location: India
Concentration: Operations
WE: Engineering (Other)

Re: What is the remainder when the number 3^1989 is divided by 7 [#permalink]
Show Tags
06 May 2016, 06:05
I have done this using binomial. 3^1989= 3.3^1988
Leave 3 aside for the moment. now, 3^1988= ((3^2))^994. = 9^994 = (7+2)^994 All the terms in the expression will be divisible by 7 except last one which is 2. So we get here 2. Now we get, 3*2/7(i had kept 3 aside in the beginning) Hence, the remainder 6.



Board of Directors
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 3326
Location: India
GPA: 3.5
WE: Business Development (Commercial Banking)

Re: What is the remainder when the number 3^1989 is divided by 7 [#permalink]
Show Tags
06 May 2016, 07:17
tallyho_88 wrote: I have done this using binomial. 3^1989= 3.3^1988
Leave 3 aside for the moment. now, 3^1988= ((3^2))^994. = 9^994 = (7+2)^994 All the terms in the expression will be divisible by 7 except last one which is 2. So we get here 2. Now we get, 3*2/7(i had kept 3 aside in the beginning) Hence, the remainder 6. No issues with your way, you might consider this as a possible way of doing the problem as well  \(3^{1989}\)= \(3^{663}\) = \(3^{3*221}\) \(\frac{3^3}{7}\)= \(\frac{27}{7}\) =6 So, \(\frac{3^{663}}{7}\) = Remainder 6 Hence answer will be C. 6
_________________
Thanks and Regards
Abhishek....
PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS
How to use Search Function in GMAT Club  Rules for Posting in QA forum  Writing Mathematical Formulas Rules for Posting in VA forum  Request Expert's Reply ( VA Forum Only )



NonHuman User
Joined: 09 Sep 2013
Posts: 13810

Re: What is the remainder when the number 3^1989 is divided by 7 [#permalink]
Show Tags
22 May 2017, 01:25
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



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7944
Location: Pune, India

Re: What is the remainder when the number 3^1989 is divided by 7 [#permalink]
Show Tags
22 May 2017, 04:44
SrinathVangala wrote: What is the remainder when the number 3^1989 is divided by 7?
A. 1 B. 5 C. 6 D. 4 E. 3 Use the concepts of Binomial theorem and negative remainders: \(3^{1989} = 3^{3*663} = 27^{663} = (28  1)^{663}\) When we use binomial to open this, we will get all terms with 28 (which is divisible by 7) except that last term which will be \((1)^{663} = 1\) So the remainder will be 1 which is the same as 7  1 = 6 (using the concept of negative remainders) For more on both these concepts, check: https://www.veritasprep.com/blog/2011/0 ... ekinyou/https://www.veritasprep.com/blog/2014/0 ... thegmat/
_________________
Karishma Veritas Prep  GMAT Instructor My Blog
Get started with Veritas Prep GMAT On Demand for $199
Veritas Prep Reviews




Re: What is the remainder when the number 3^1989 is divided by 7
[#permalink]
22 May 2017, 04:44






