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

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Re: What is the remainder when 52^60 is divided by 31 [#permalink]
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...
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Re: What is the remainder when 52^60 is divided by 31 [#permalink]
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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
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Re: What is the remainder when 52^60 is divided by 31 [#permalink]
Great, thank you Kinshook!
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Re: What is the remainder when 52^60 is divided by 31 [#permalink]
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Re: What is the remainder when 52^60 is divided by 31 [#permalink]
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