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

 It is currently 25 Jun 2019, 13:24 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

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  Remainder when 2^100 is divided by 9

Author Message
TAGS:

Hide Tags

Manager  S
Joined: 12 Dec 2017
Posts: 79
Location: India
Schools: Yale '19, GMBA '20, XLRI
GMAT 1: 660 Q46 V35 GPA: 3.8
Remainder when 2^100 is divided by 9  [#permalink]

Show Tags

1 00:00

Difficulty:   45% (medium)

Question Stats: 61% (01:33) correct 39% (01:28) wrong based on 62 sessions

HideShow timer Statistics What is the remainder when 2^100 is divided by 9?

a. 1
b. 3
c. 4
d. 7
e. 8

Originally posted by kanakdaga on 10 Jan 2019, 08:29.
Last edited by kanakdaga on 10 Jan 2019, 08:42, edited 1 time in total.
Intern  B
Joined: 23 Feb 2012
Posts: 42
Re: Remainder when 2^100 is divided by 9  [#permalink]

Show Tags

1
1
2^6=64 ND 9*7=63 so remainder is 1 for 2^96 is 1*2^4 = 16 divided by 9...remainder 7

Sent from my Redmi 5 Plus using GMAT Club Forum mobile app
Intern  B
Joined: 15 Nov 2018
Posts: 34
Location: United States
Concentration: Operations, Finance
GMAT 1: 650 Q48 V31 GPA: 3
WE: Engineering (Other)
Re: Remainder when 2^100 is divided by 9  [#permalink]

Show Tags

1
I think the OA is wrong

Posted from my mobile device
Manager  S
Joined: 12 Dec 2017
Posts: 79
Location: India
Schools: Yale '19, GMBA '20, XLRI
GMAT 1: 660 Q46 V35 GPA: 3.8
Re: Remainder when 2^100 is divided by 9  [#permalink]

Show Tags

deepverma wrote:
2^6=64 ND 9*7=63 so remainder is 1 for 2^96 is 1*2^4 = 16 divided by 9...remainder 7

Sent from my Redmi 5 Plus using GMAT Club Forum mobile app

couldn't really understand! can you elaborate ?
Manager  S
Joined: 12 Dec 2017
Posts: 79
Location: India
Schools: Yale '19, GMBA '20, XLRI
GMAT 1: 660 Q46 V35 GPA: 3.8
Re: Remainder when 2^100 is divided by 9  [#permalink]

Show Tags

rajatvermaenator wrote:
I think the OA is wrong

Posted from my mobile device

apologies!
Corrected now!
Manager  S
Joined: 12 Dec 2017
Posts: 79
Location: India
Schools: Yale '19, GMBA '20, XLRI
GMAT 1: 660 Q46 V35 GPA: 3.8
Re: Remainder when 2^100 is divided by 9  [#permalink]

Show Tags

2
Please check if my approach is correct!

9=8+1
9= 2^3 + 1
or 2^3 = 9 - 1

When there are 100 2s and we need groups of 3 2s , it will be possible by the following way:

2^(99) * 2^(1) = 2^(100)
2^(3*33) * 2^(1) = 2^(100)
8^(33) * 2^(1) = 2^(100)
(9-1)^(33) * 2^(1) = 2^(100)

9 will leave remainder 0
-1^33 will leave remainder -1
2^1 will leave remainder 2

Thus , -1 * 2 = -2

Now we add divisor to negative remainder : -2 + 9 = 7

Hence, 7 is the actual remainder.

KUDOS IF THIS HELPS! :D
Intern  B
Joined: 10 Jan 2014
Posts: 6
GPA: 3.6
Re: Remainder when 2^100 is divided by 9  [#permalink]

Show Tags

1
2^1=2
2^2=4
2^3=8
2^4=16
2^5=32
thus the cyclicity is 4m+1, 4m+2, 4m+3, 4m
2^100=2^4*25=2^4m, which is the same as 2^4=16=2^4*4=2^4m.

thus I simply use 16 divided by 9 and its remainder is 7, so D
Senior Manager  S
Joined: 12 Sep 2017
Posts: 274
Remainder when 2^100 is divided by 9  [#permalink]

Show Tags

Hello!

Can someone please provide a short cut for this problem?

I took a lot of time to understand this one:

2/9 = 2
4/9 = 4
8/9 = 8
16/9 = 7
32/9 = 5
64/9 = 1
128/9 = 2
256/9 = 4
512/9 = 8
1024/9 = 7
../9 = 5
.../9 = 1

Hence the cyclicity of remainders is 2,4,8,7,5,1.

Since 100 is even then the cyclicity for evens is 4,7,1.

From here I've got a doubt, ¿what should I choose?

I took 7 cuz 2 power 100 is similar to 2 power 10, which holds a remainder of 7.

Kind regards!
VP  G
Joined: 09 Mar 2018
Posts: 1003
Location: India
Remainder when 2^100 is divided by 9  [#permalink]

Show Tags

jfranciscocuencag wrote:
Hello!

Can someone please provide a short cut for this problem?

I took a lot of time to understand this one:

2/9 = 2
4/9 = 4
8/9 = 8
16/9 = 7
32/9 = 5
64/9 = 1
128/9 = 2
256/9 = 4
512/9 = 8
1024/9 = 7
../9 = 5
.../9 = 1

Hence the cyclicity of remainders is 2,4,8,7,5,1.

Since 100 is even then the cyclicity for evens is 4,7,1. - This was not required as such

From here I've got a doubt, ¿what should I choose?

I took 7 cuz 2 power 100 is similar to 2 power 10, which holds a remainder of 7.

Kind regards!

Hi, Actually you were on the right track, and got the correct sequence of cyclicity

Just that you didnt have to check for even numbers as such

cyclicity of remainders is 2,4,8,7,5,1

So if you notice that its a series of 6 numbers, which will be repeated all the time

16*6 =96 this means for 2^96 the remainder will be 1, after this move 4 times, in the series of cyclcity, To give you the position for 2^100.

Kindly let me know if this helps you.
_________________
If you notice any discrepancy in my reasoning, please let me know. Lets improve together.

Quote which i can relate to.
Many of life's failures happen with people who do not realize how close they were to success when they gave up.
Senior Manager  S
Joined: 12 Sep 2017
Posts: 274
Re: Remainder when 2^100 is divided by 9  [#permalink]

Show Tags

1
KanishkM wrote:
jfranciscocuencag wrote:
Hello!

Can someone please provide a short cut for this problem?

I took a lot of time to understand this one:

2/9 = 2
4/9 = 4
8/9 = 8
16/9 = 7
32/9 = 5
64/9 = 1
128/9 = 2
256/9 = 4
512/9 = 8
1024/9 = 7
../9 = 5
.../9 = 1

Hence the cyclicity of remainders is 2,4,8,7,5,1.

Since 100 is even then the cyclicity for evens is 4,7,1. - This was not required as such

From here I've got a doubt, ¿what should I choose?

I took 7 cuz 2 power 100 is similar to 2 power 10, which holds a remainder of 7.

Kind regards!

Hi, Actually you were on the right track, and got the correct sequence of cyclicity

Just that you didnt have to check for even numbers as such

cyclicity of remainders is 2,4,8,7,5,1

So if you notice that its a series of 6 numbers, which will be repeated all the time

16*6 =96 this means for 2^96 the remainder will be 1, after this move 4 times, in the series of cyclcity, To give you the position for 2^100.

Kindly let me know if this helps you.

Now its clear for me, thank you!!! Re: Remainder when 2^100 is divided by 9   [#permalink] 12 Jan 2019, 15:29
Display posts from previous: Sort by

Remainder when 2^100 is divided by 9  