Finding the last two digits

Question

How can I determine the last two digits of the decimal representation?

My example is 17^362.

Thanks for help!

Answer 1

Determining the last two digits of a positive integer is equivalent to calculating that integer modulo 100. By the Chinese remainder theorem, it suffices to calculate $17^{362}$ modulo $4$ and modulo $25$. We see that $17 \equiv 1 \bmod 4$, so $17^{362} \equiv 1 \bmod 4$ as well. Working modulo 25, we have by Euler's theorem that $1 \equiv 17^{\phi(25)} = 17^{20} \pmod {25}$. It follows that $17^{360} \equiv 1$, so $$17^{362} \equiv 17^2 \equiv 14 \pmod {25}$$ We have now found that $17^{362} \equiv 1 \pmod 4$ and $17^{362} \equiv 14 \pmod {25}$, so we conclude by the CRT that $17^{362} \equiv 89 \pmod {100}$. This means that the last two digits of $17^{362}$ are $89$.