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What is the remainder of 5^{2010} when it is divided by 7?

 Aug 12, 2018
 #1
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Any power of 5 in the form of 6n mod 7 =1, where n=1, 2, 3......etc.

Since 2010 / 6 =335, it, therefore, follows that 5^(6*335) mod 7 = 1

 Aug 13, 2018
 #2
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What is the remainder of 5^{2010} when it is divided by 7?

 

\(5^{2010}\:mod\;7\\ \equiv (-2)^{2010}\:mod\;7\\ \equiv2^{2010}\:mod\;7\\ \equiv 2^{3*670}\:mod\;7\\ \equiv 8^{670}\:mod\;7\\ \equiv 1^{670}\:mod\;7\\ \equiv 1\:mod\;7\\\)

 

so the remainder is 1

 Aug 18, 2018

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