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?
Number Theory – Finding the Last Two Digits of 6543^210
elementary-number-theorymodular arithmetic
Best Answer
One can use the binomial theorem (thrice). To do so, write $h$ for everything that is a multiple of $100$, possibly varying from line to line.
Since $6543=43+h$, one knows that $$ 6543^{210}=(43+h)^{210}=43^{210}+h. $$ Now, $$ 43^{210}=(3+40)^{210}=3^{210}+210\cdot3^{209}\cdot40+h=3^{210}+h. $$ Finally, $$ 3^{210}=9^{105}=(-1+10)^{105}=-1+105\cdot10+h=1049+h=49+h, $$ that is, $6543^{210}=49+$ some multiple of $100$.