Summer is Coming! Join the Game of Timers Competition to Win Epic Prizes. Registration is Open. Game starts Mon July 1st.

 It is currently 19 Jul 2019, 16:05

### 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 hundreds digit of 201^201

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Aug 2009
Posts: 7764
What is the hundreds digit of 201^201  [#permalink]

### Show Tags

13 Aug 2018, 20:05
00:00

Difficulty:

75% (hard)

Question Stats:

65% (01:48) correct 35% (01:42) wrong based on 122 sessions

### HideShow timer Statistics

What is the hundreds digit of $$201^{201}$$ ?
(A) 0
(B) 1
(C) 2
(D) 5
(E) 7

New question!!!..

_________________
Math Expert
Joined: 02 Aug 2009
Posts: 7764
What is the hundreds digit of 201^201  [#permalink]

### Show Tags

13 Aug 2018, 20:55
chetan2u wrote:
What is the hundreds digit of $$201^{201}$$ ?
(A) 0
(B) 1
(C) 2
(D) 5
(E) 7

New question!!!..

Apart from cyclicity, we can read this as remainder when divided by 1000..
so $$201^{201}=(200+1)^{201}=200^{201}+.....200^2*1^{199}+200^1*1^200+1^{201}$$
so all terms apart from last two terms will be divisible by 1000..
hence remainder will be $$200^1*1+1=201$$
the hundreds digit therefore will be 2

C
_________________
VP
Status: Learning stage
Joined: 01 Oct 2017
Posts: 1028
WE: Supply Chain Management (Energy and Utilities)
Re: What is the hundreds digit of 201^201  [#permalink]

### Show Tags

Updated on: 13 Aug 2018, 21:24
chetan2u wrote:
chetan2u wrote:
What is the hundreds digit of $$201^{201}$$ ?
(A) 0
(B) 1
(C) 2
(D) 5
(E) 7

New question!!!..

pattern of hundreds digit of $$(201)^n$$:-
2-4-6-8-0
Cyclicity:- 5

We have $$\frac{201}{5}$$ leaves 1 as remainder.
So, hundreds digit of $$(201)^{201}$$ is same as hundreds digit of $$(201)^1$$.

Therefore, the required hundreds digit is 2.

Ans. (C)
_________________
Regards,

PKN

Rise above the storm, you will find the sunshine

Originally posted by PKN on 13 Aug 2018, 21:08.
Last edited by PKN on 13 Aug 2018, 21:24, edited 1 time in total.
Senior PS Moderator
Joined: 26 Feb 2016
Posts: 3359
Location: India
GPA: 3.12
Re: What is the hundreds digit of 201^201  [#permalink]

### Show Tags

13 Aug 2018, 21:20
1
chetan2u wrote:
What is the hundreds digit of $$201^{201}$$ ?
(A) 0
(B) 1
(C) 2
(D) 5
(E) 7

New question!!!..

$$(201)^0$$ - Hundred's digit is 0
$$(201)^1$$ - Hundred's digit is 2
$$(201)^2$$ - Hundred's digit is 4
$$(201)^3$$ - Hundred's digit is 6
$$(201)^4$$ - Hundred's digit is 8
$$(201)^5$$ - Hundred's digit is 0
$$(201)^6$$ - Hundred's digit is 2
$$(201)^7$$ - Hundred's digit is 4

A pattern emerges and 8 is the hundred's digit for the 4,9,14,19.......199th power of 201.

Therefore, the hundred's digit of $$201^{201}$$ is 2(Option C)
_________________
You've got what it takes, but it will take everything you've got
Senior Manager
Joined: 15 Feb 2017
Posts: 301
Re: What is the hundreds digit of 201^201  [#permalink]

### Show Tags

13 Aug 2018, 21:37
[quote="chetan2u"]What is the hundreds digit of $$201^{201}$$ ?
(A) 0
(B) 1
(C) 2
(D) 5
(E) 7

Hundreds digit is 2.please correct if I am wrong

Sent from my Lenovo K33a42 using GMAT Club Forum mobile app
_________________
IMPOSSIBLE IS JUST AN OPINION
Manager
Joined: 20 Apr 2018
Posts: 173
Concentration: Technology, Nonprofit
WE: Analyst (Non-Profit and Government)
Re: What is the hundreds digit of 201^201  [#permalink]

### Show Tags

13 Aug 2018, 22:42
1
PKN wrote:
chetan2u wrote:
chetan2u wrote:
What is the hundreds digit of $$201^{201}$$ ?
(A) 0
(B) 1
(C) 2
(D) 5
(E) 7

New question!!!..

pattern of hundreds digit of $$(201)^n$$:-
2-4-6-8-0
Cyclicity:- 5

PKN, how do you know the pattern is 2-4-6-8-0?

I solved this using binomial expansion.
If you think of 201^201 = (200+1)^201 = 201C0*200^201 + 201C1*200^200 + ... + 201C200*200^1 + 201C201*200^0
^the highlighted term equal 201*200 = 40200
^the highlighted part is the hundreds digit. All other terms will have higher powers and won't contribute to the hundreds term.

VP
Status: Learning stage
Joined: 01 Oct 2017
Posts: 1028
WE: Supply Chain Management (Energy and Utilities)
Re: What is the hundreds digit of 201^201  [#permalink]

### Show Tags

13 Aug 2018, 23:07
3
sandman13 wrote:
PKN wrote:

pattern of hundreds digit of $$(201)^n$$:-
2-4-6-8-0
Cyclicity:- 5

PKN, how do you know the pattern is 2-4-6-8-0?

I solved this using binomial expansion.
If you think of 201^201 = (200+1)^201 = 201C0*200^201 + 201C1*200^200 + ... + 201C200*200^1 + 201C201*200^0
^the highlighted term equal 201*200 = 40200
^the highlighted part is the hundreds digit. All other terms will have higher powers and won't contribute to the hundreds term.

Hi sandman13,
If you observe carefully, We have each of the digit of 201 , lies between 0 and 2 inclusive. On successive powers of 201 from 1 to infinity, the tens digit of 201 decides the hundreds digit(1*any number=that number.no carry forward)). Here the tens digit is zero, hence 0*any number=0 with no carry forward.

Hundreds digit of $$201^1$$=2
Hundreds digit of $$201^2$$= 2+Hundreds digit of $$201^1=2+2=4$$
Hundreds digit of $$201^3$$= 2+Hundreds digit of $$201^2=2+4=6$$
Hundreds digit of $$201^4$$= 2+Hundreds digit of $$201^3=2+6=8$$
Hundreds digit of $$201^5$$= 2+Hundreds digit of $$201^4=2+8=0$$
Hundreds digit of $$201^6$$= 2+Hundreds digit of $$201^5=2+0=2$$
Hundreds digit of $$201^7$$= 2+Hundreds digit of $$201^6=2+2=4$$

This was my approach, it mayn't be a standard one.
_________________
Regards,

PKN

Rise above the storm, you will find the sunshine
Intern
Joined: 14 Jan 2018
Posts: 45
Location: India
Concentration: General Management, Entrepreneurship
GMAT 1: 660 Q50 V29
GPA: 3.8
WE: Engineering (Manufacturing)
Re: What is the hundreds digit of 201^201  [#permalink]

### Show Tags

13 Aug 2018, 23:27
pattern of hundreds digit of (201)^(201):-
2-4-6-8-0
Cyclicity:- 5
So 201/5 remainder =1 so the hundred digit will be 2

Option C

Posted from my mobile device
LBS Moderator
Joined: 04 Jun 2018
Posts: 587
Location: Germany
Concentration: General Management, Finance
GMAT 1: 730 Q47 V44
GPA: 3.4
WE: Analyst (Transportation)
Re: What is the hundreds digit of 201^201  [#permalink]

### Show Tags

20 Aug 2018, 17:36
chetan2u wrote:
chetan2u wrote:
What is the hundreds digit of $$201^{201}$$ ?
(A) 0
(B) 1
(C) 2
(D) 5
(E) 7

New question!!!..

Post for OE

Is the OA really correct? Everybody seems to chose and provide explanations for C...

Incase C is incorrect, could someone provide an explanation as to why C is wrong?

Regards,
Chris
_________________
Intern
Joined: 21 Jan 2017
Posts: 32
What is the hundreds digit of 201^201  [#permalink]

### Show Tags

21 Aug 2018, 04:58
chetan2u wrote:
What is the hundreds digit of $$201^{201}$$ ?
(A) 0
(B) 1
(C) 2
(D) 5
(E) 7

New question!!!..

My approach might be lengthy yet easy! Here it goes.

Note: 401, 601 etc are all the last 3 digits

$$201^{201}$$ can be written as $$201^{200}$$ * 201.
= $$(201^{2})^{100}$$ * 201
= $$(401)^{100}$$ * 201
= $$(401^{2})^{50}$$ * 201
= $$801^{50}$$ * 201
= $$((801)^{2})^{25}$$ * 201
= $$(601)^{25}$$ *201
= $$(601^{2})^{12}$$ * 201 * 601
= $$(201)^{12}$$ *201 * 601
= $$((201)^{2})^{6}$$ *201 *601

again $$201^2$$ is 401 hence substitute,
$$401^2$$ is 801
= 601 * 801 * 201 * 601
the hundreds digit of the above is 2.

Thanks,
Uma
VP
Status: Learning stage
Joined: 01 Oct 2017
Posts: 1028
WE: Supply Chain Management (Energy and Utilities)
Re: What is the hundreds digit of 201^201  [#permalink]

### Show Tags

21 Aug 2018, 08:34
Arro44 wrote:
chetan2u wrote:
chetan2u wrote:
What is the hundreds digit of $$201^{201}$$ ?
(A) 0
(B) 1
(C) 2
(D) 5
(E) 7

New question!!!..

Post for OE

Is the OA really correct? Everybody seems to chose and provide explanations for C...

Incase C is incorrect, could someone provide an explanation as to why C is wrong?

Regards,
Chris

Hi Arro44,
OA is (C).
Do you have any points?

Posted from my mobile device
_________________
Regards,

PKN

Rise above the storm, you will find the sunshine
Re: What is the hundreds digit of 201^201   [#permalink] 21 Aug 2018, 08:34
Display posts from previous: Sort by