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

 It is currently 03 Aug 2020, 03:57

### 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

# What is the remainder when 2^1344452457 is divided by 11?

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 65761
What is the remainder when 2^1344452457 is divided by 11?  [#permalink]

### Show Tags

19 Sep 2019, 06:03
1
10
00:00

Difficulty:

55% (hard)

Question Stats:

55% (01:55) correct 45% (01:23) wrong based on 92 sessions

### HideShow timer Statistics

What is the remainder when $$2^{1344452457}$$ is divided by 11?

A. 2
B. 4
C. 5
D. 7
E. 9

_________________
SVP
Joined: 20 Jul 2017
Posts: 1504
Location: India
Concentration: Entrepreneurship, Marketing
WE: Education (Education)
Re: What is the remainder when 2^1344452457 is divided by 11?  [#permalink]

### Show Tags

19 Sep 2019, 06:22
6
2
Bunuel wrote:
What is the remainder when $$2^{1344452457}$$ is divided by 11?

A. 2
B. 4
C. 5
D. 7
E. 9

$$2^{1344452457}$$

We can write 1344452457 = 5k + 2, for some odd integer k
--> $$2^{1344452457}$$
--> $$2^{5k + 2}$$
--> 4*$$2^{5k}$$
--> 4*$$32^k$$
--> 4*$$(33 - 1)^k$$

Since, k is an odd integer $$(33 - 1)^k$$ will be of the form 11A - 1 or 11B + 10, for some positive integers A or B
-> 4*$$(33 - 1)^k$$ = 4*(11B + 10) = 44B + 40

Remainder when 44B + 40 is divided by 11 is 7 [Since, 44B + 40 = 11(4B + 3) + 7]

IMO Option D

Pls Hit Kudos if you like the solution
##### General Discussion
GMAT Tutor
Joined: 16 Sep 2014
Posts: 554
Location: United States
GMAT 1: 780 Q51 V45
GRE 1: Q170 V167
What is the remainder when 2^1344452457 is divided by 11?  [#permalink]

### Show Tags

Updated on: 19 Sep 2019, 06:34
We would like to use the theorem $$x^n$$ % 11 = $$(x - 11i)^n$$ % 11 where $$i$$ is any integer and make $$x - 11i$$ either 1 or -1 to evaluate the remainder (x % 11 means the remainder of x/11). To do that we must change the base of the exponent close to a multiple of 11. The closest one, for now, is $$2^5 = 32$$. So let us try to change the base to 32 and subtract 33.

$$2^{1344452457} = 2^2 * 2^{1344452455} = 2^2 * 2^{5 * n} = 2^2 * 32^n$$. Here n represent the result of 1344452455/5 which is trivial.

One thing to note here is that n must be odd since the original exponent ends in 5. Then we can do the following:

$$2^2 * \frac{{32^n}}{11} = 4 * \frac{{32^n}}{11}$$

If we focus on $$\frac{{32^n}}{11}$$ we have:

$$32^n$$ % 11 = $$(-1)^n$$ % 11 = -1 % 11 (since n is odd) = -1.

Finally we multiply that by 4 to get -4 is the remainder, or -4 + 11 = 7 is the remainder. The answer is D.
_________________
Source: We are an NYC based, in-person and online GMAT tutoring and prep company. We are the only GMAT provider in the world to guarantee specific GMAT scores with our flat-fee tutoring packages, or to publish student score increase rates. Our typical new-to-GMAT student score increase rate is 3-9 points per tutoring hour, the fastest in the world. Feel free to reach out!

Originally posted by TestPrepUnlimited on 19 Sep 2019, 06:14.
Last edited by TestPrepUnlimited on 19 Sep 2019, 06:34, edited 1 time in total.
Director
Joined: 08 Aug 2017
Posts: 762
Re: What is the remainder when 2^1344452457 is divided by 11?  [#permalink]

### Show Tags

19 Sep 2019, 08:14
I wrote 1344452457= 4K+1
So it becomes 2*2^(4*k)= 2*16^k
Reminder(16^K/11) = 5
And reminder of 2/11= 2
Therefore, 5*2= 10

Where I am wrong?
DS Forum Moderator
Joined: 19 Oct 2018
Posts: 2047
Location: India
Re: What is the remainder when 2^1344452457 is divided by 11?  [#permalink]

### Show Tags

19 Sep 2019, 08:35
2
$$2^5$$= -1 mod 11
$$(2^5)^{odd}$$= $$(-1)^{odd}$$ mod 11

$$(2^5)^{odd}$$*$$2^2$$= -1*4 mod 11

1344452457= 5*odd+2
$$2^{1344452457}$$=-4 mod 11=7 mod 11

Bunuel wrote:
What is the remainder when $$2^{1344452457}$$ is divided by 11?

A. 2
B. 4
C. 5
D. 7
E. 9
Intern
Joined: 14 Sep 2014
Posts: 14
Location: India
Schools: IIMA PGPX "21
Re: What is the remainder when 2^1344452457 is divided by 11?  [#permalink]

### Show Tags

19 Sep 2019, 09:55
2
1
Bunuel wrote:
What is the remainder when $$2^{1344452457}$$ is divided by 11?

A. 2
B. 4
C. 5
D. 7
E. 9

2^1 % 11 = 2
2^2 % 11 = 4
-
-
-
-
2^10 = 1024 % 11 = 1

make power a multiple of 10 so we have 2^(10K + 7 )

Remainder from 2^10K = 1
remainder from 2^7 = 7
Senior Manager
Status: Whatever it takes!
Joined: 10 Oct 2018
Posts: 372
GPA: 4
What is the remainder when 2^1344452457 is divided by 11?  [#permalink]

### Show Tags

19 Sep 2019, 10:23
Bunuel wrote:
What is the remainder when $$2^{1344452457}$$ is divided by 11?

A. 2
B. 4
C. 5
D. 7
E. 9

The easiest and quickest way to solve this question is by pattern by using the formula=> dividend = divisor x quotient + remainder
2^1= 11(0) + 2 =>r=2
2^2= 11(0) + 4 =>r=4
2^3= 11(0) + 8 =>r=8
2^4= 11(1) + 5 =>r=5
2^5= 11(2) + 10 =>r=10
2^6= 11(5) + 9 =>r=9
2^7= 11(11) + 7 =>r=7
2^8= 11(23) + 3=>r=3
2^9= 11(46)+6=>r=6
2^10= 11(93)+1=>r=1
2^11= 11(186)+2=>r=2
2^12= 11(372)+4=>r=4
....
....
....
Notice the cyclicity of 10 starts at 2^11.....which means remainder for 2^(any number ending with 7) will be 7. Hence, option D

P.S: This method might seem lengthy. I am aware of the values till 2^10 which made my calculations easier. If you are quick at calculations (calculation with 2 is very quick and simple) and approximation, then this method will be the best choice (at a point I was really pissed off while calculating cyclicity....if this happened in exam, I would have chosen an remainder of 2^7 and moved on -_- )
Intern
Joined: 26 May 2014
Posts: 15
Re: What is the remainder when 2^1344452457 is divided by 11?  [#permalink]

### Show Tags

20 Jun 2020, 19:19
Quick way is the use of Totient formula for finding out the remainder.
Re: What is the remainder when 2^1344452457 is divided by 11?   [#permalink] 20 Jun 2020, 19:19