TITLE: Finding the last two digits of $6543^{210}$
QUESTION [6 upvotes]: I have to find the last two digits of $6543^{210}$, my strategy is to use the Euler theorem and then some algebra to reduce this to $6543^{10}$, however I can't think of any easy way to proceed after this, any ideas?

REPLY [5 votes]: Binomial Theorem $\Rightarrow\rm mod\ 100\!:\ (-7\!+\!50)^{\large 2+4N}\!\equiv (-7)^{\large 2+4N}\! \equiv 7^{\large 2}\, $ by $\, 7^{\large 4}\! \equiv (-1\!+\!50)^{\large 2}\!\equiv 1$