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What is the remainder when 52^60 is divided by 31

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Math Expert
Joined: 02 Sep 2009
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What is the remainder when 52^60 is divided by 31  [#permalink]

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24 Feb 2020, 04:30
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Difficulty:

55% (hard)

Question Stats:

35% (01:28) correct 65% (02:12) wrong based on 26 sessions

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What is the remainder when $$52^{60}$$ is divided by 31 ?

A. 0
B. 1
C. 21
D. 23
E. 30

Are You Up For the Challenge: 700 Level Questions

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Joined: 02 Aug 2009
Posts: 8295
Re: What is the remainder when 52^60 is divided by 31  [#permalink]

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24 Feb 2020, 06:35
2
1
Bunuel wrote:
What is the remainder when $$52^{60}$$ is divided by 31 ?

A. 0
B. 1
C. 21
D. 23
E. 30

Are You Up For the Challenge: 700 Level Questions

Working out a pattern will be cumbersome, so let us do by BINOMIAL theorem..

$$52^{60}=(62-10)^{60}$$, so remainder will be $$(-10)^{60}=100^{30}=(93+7)^{30}$$..
Remainder when $$(93+7)^{30}$$ is divided by 31 will be $$7^{30}=(7^3)^{10}=343^{10}=(341+2)^{10}=(31*11+2)^{10}$$
Remainder when $$(31*11+2)^{10}$$ is divided by 31 will be$$2^{10}=(2^5)^2=32^2=(31+1)^2$$
Thus the remainder = $$1^2$$ or 1

B
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What is the remainder when 52^60 is divided by 31  [#permalink]

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29 Mar 2020, 02:11
2
Bunuel wrote:
What is the remainder when $$52^{60}$$ is divided by 31 ?

A. 0
B. 1
C. 21
D. 23
E. 30

Are You Up For the Challenge: 700 Level Questions

52 = 62 - 10

52^60mod31 = (-10)^60mod31 = 10^60mod31 = 7^30mod31 = 2^10mod31 = 1^2mod31 = 1

IMO B
Intern
Joined: 22 Mar 2019
Posts: 2
Re: What is the remainder when 52^60 is divided by 31  [#permalink]

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29 Mar 2020, 05:07
Quote:
52 = 62 - 10

52^60mod31 = (-10)^60mod31 = 10^60mod31 = 7^30mod31 = 2^10mod31 = 1^2mod31 = 1

IMO B

Kinshook Could you please explain what you've exactly done in this part of your solution "10^60mod31 = 7^30mod31 = 2^10mod31 = 1^2mod31"?

I understand this part "52^60mod31 = (-10)^60mod31 = 10^60mod31" but then I would have proceded the following way: 2^60*5^60mod31 = 32^12*125^20mod31 = 1^12*1^20mod31 = 1. However, it took me too much time to come up with my solution, it seems to me that your solution is much faster and that you know some tricks in modular arithmetic that I don't know yet...
CEO
Joined: 03 Jun 2019
Posts: 2500
Location: India
GMAT 1: 690 Q50 V34
WE: Engineering (Transportation)
Re: What is the remainder when 52^60 is divided by 31  [#permalink]

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29 Mar 2020, 05:15
1
Vendap02 wrote:
Quote:
52 = 62 - 10

52^60mod31 = (-10)^60mod31 = 10^60mod31 = 7^30mod31 = 2^10mod31 = 1^2mod31 = 1

IMO B

Kinshook Could you please explain what you've exactly done in this part of your solution "10^60mod31 = 7^30mod31 = 2^10mod31 = 1^2mod31"?

I understand this part "52^60mod31 = (-10)^60mod31 = 10^60mod31" but then I would have proceded the following way: 2^60*5^60mod31 = 32^12*125^20mod31 = 1^12*1^20mod31 = 1. However, it took me too much time to come up with my solution, it seems to me that your solution is much faster and that you know some tricks in modular arithmetic that I don't know yet...

Hi Vendap02
10^60mod31 = 100^30mod31 = (93+7)^30mod31 = 7^30mod31 = = 343^10mod31 = (341+2)^10mod31 = 2^10mod31 = (31+1)^2mod31 = 1^2mod31 = 1
Hope it is clear now
Intern
Joined: 22 Mar 2019
Posts: 2
Re: What is the remainder when 52^60 is divided by 31  [#permalink]

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29 Mar 2020, 05:21
Great, thank you Kinshook!
Re: What is the remainder when 52^60 is divided by 31   [#permalink] 29 Mar 2020, 05:21
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