Need examples of $n^{20}$ & $n^{200}$ to see a pattern.

Question

Let n be an even integer not divisible by 10. What digit will be in the tens place in $n^{20}$, and in the hundreds place in $n^{200}$? Can you generalise this?

The problem I have is that I can't seem to find examples to see some pattern. A Casio scientific calculator can only go up to $4^{20}$, and Google isn't giving me much either. So I don't know what I'm supposed to see, let alone how to generalise this.

Can you please help me?

Edit 1: Turns out I didn't understand the question. I'm supposed to look at the tens (ie. 2nd) digit of $n^{20}$ and the hundreds (ie. 3rd) digit of $n^{200}$. As I'm recording examples from $n=2, 4, 6, 8, 12, 14, 16, 18,...$, I still don't see a pattern. Is there supposed to be a pattern, and I'm just doing it wrong, or nuh?

Edit 2: I didn't see a pattern because I was looking at the front digits rather than the back digits (Hi GCSE maths! It's been a minute!). But this brings me back to the very first question I asked. While $2^{20} = 1048576$, $2^{200} = 1.606938e+60$. How do I find the hundreds digit from that?

Answer 1

Take for example the number $2$ so you will get $$2^{20}=1048576$$ and $$2^{100}=1267650600228229401496703205376$$

Answer 2

I think what this question is really about is proving $100|n^{200}-n^{20}$. It will help to first prove $100|n^{20}-k$ for some $k\in\{0,\,1,\,25,\,76\}$, and that $100|k^{10}-k$ for each such $k$. You can do the first part by considering $n^{20}$ modulo $4,\,25$ with the Fermat-Euler theorem; note that $\varphi(4)=2,\,\varphi(25)=20$ both divide $20$.

Answer 3

I expect what they're hoping for you to discover is that, for $n\neq 0\pmod{5}$, $n=0\pmod{2}$, $$\begin{align} n^{20} &= ? \pmod{100} \\ n^{200} &= ? \pmod{1000} \\ n^{2000} &= 9376 \pmod{10000} \\ n^{20000} &= 09376 \pmod{100000} \\ n^{200000} &= 109376 \pmod{1000000} \\ n^{2000000} &= 7109376 \pmod{10000000}\text{.} \end{align}$$ These results follow from the binomial theorem, Euler's generalization of Fermat's little theorem, and the Chinese remainder theorem.