What is the remainder of 5^{2010} when it is divided by 7?
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
\(5^{2010}\:mod\;7\\ \equiv (-2)^{2010}\:mod\;7\\ \equiv2^{2010}\:mod\;7\\ \equiv 2^{3*670}\:mod\;7\\ \equiv 8^{670}\:mod\;7\\ \equiv 1^{670}\:mod\;7\\ \equiv 1\:mod\;7\\\)
so the remainder is 1
