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

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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
+102441
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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