GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 15 Feb 2019, 11:57

### 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

## Events & Promotions

###### Events & Promotions in February
PrevNext
SuMoTuWeThFrSa
272829303112
3456789
10111213141516
17181920212223
242526272812
Open Detailed Calendar
• ### Free GMAT practice

February 15, 2019

February 15, 2019

10:00 PM EST

11:00 PM PST

Instead of wasting 3 months solving 5,000+ random GMAT questions, focus on just the 1,500 you need.

# What is the remainder when the number 3^1989 is divided by 7

Author Message
TAGS:

### Hide Tags

Intern
Joined: 05 Mar 2013
Posts: 44
Location: India
Concentration: Entrepreneurship, Marketing
GMAT Date: 06-05-2013
GPA: 3.2
What is the remainder when the number 3^1989 is divided by 7  [#permalink]

### Show Tags

Updated on: 17 May 2013, 16:19
4
34
00:00

Difficulty:

85% (hard)

Question Stats:

53% (02:00) correct 47% (02:01) wrong based on 350 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

_________________

"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.

Originally posted by SrinathVangala on 17 May 2013, 06:43.
Last edited by Bunuel on 17 May 2013, 16:19, edited 1 time in total.
Edited the question.
Math Expert
Joined: 02 Sep 2009
Posts: 52905
Re: What is the remainder when the number 3^1989 is divided by 7  [#permalink]

### Show Tags

16 Apr 2014, 04:51
8
7
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: {6-1}{6-1}{6-1}... 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.

Units digits, exponents, remainders problems: new-units-digits-exponents-remainders-problems-168569.html

Hope it helps.
_________________
Intern
Joined: 04 Jan 2013
Posts: 13
Location: India
Concentration: Finance
GMAT Date: 08-26-2013
GPA: 2.83
WE: Other (Other)
Re: What is the remainder???  [#permalink]

### Show Tags

17 May 2013, 06:48
3
2
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.
##### General Discussion
Intern
Joined: 05 Mar 2013
Posts: 44
Location: India
Concentration: Entrepreneurship, Marketing
GMAT Date: 06-05-2013
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: 121
Location: India
Concentration: General Management, Technology
GMAT 1: 640 Q42 V37
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*

Intern
Joined: 05 Apr 2010
Posts: 12
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: 08-26-2013
GPA: 2.83
WE: Other (Other)
Re: What is the remainder???  [#permalink]

### Show Tags

17 May 2013, 07:19
1
1
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.
Current Student
Joined: 06 Jan 2013
Posts: 25
GPA: 4
WE: Engineering (Transportation)
Re: What is the remainder???  [#permalink]

### Show Tags

17 May 2013, 09:09
1
1
This is how I solved it:-
$$\frac{3^{1989}}{7}=\frac{(7-4)^{1989}}{7}$$
Every term in the expansion of $$(7-4)^{1989}$$ 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: 77
Schools: ISB '15
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)

Manager
Joined: 22 Aug 2014
Posts: 149
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
6/7-6 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?
CEO
Joined: 20 Mar 2014
Posts: 2629
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44
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?

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 what-is-the-remainder-when-the-number-3-1989-is-divided-by-152951.html#p1356693

One 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.
Current Student
Joined: 15 Mar 2016
Posts: 96
Location: India
Concentration: Operations
GMAT 1: 680 Q47 V36
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: 4382
Location: India
GPA: 3.5
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 )

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8880
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
1
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 ... ek-in-you/
https://www.veritasprep.com/blog/2014/0 ... -the-gmat/
_________________

Karishma
Veritas Prep GMAT Instructor

Intern
Joined: 27 Jun 2015
Posts: 13
Re: What is the remainder when the number 3^1989 is divided by 7  [#permalink]

### Show Tags

11 Feb 2019, 19:24
Cyclicity when 3^n divided by 7 is 6, thus divided 1989 by 6 we get reminder as 3 and and when 3^3 is divided by 7 it gives reminder as 6, thus the answer will be 6

Posted from my mobile device
Re: What is the remainder when the number 3^1989 is divided by 7   [#permalink] 11 Feb 2019, 19:24
Display posts from previous: Sort by