Last Digits Curious Observation

Question

I have made the following observations and I need its detailed proof or, if wrong, the disproof.

Consider a positive integer $ 'a' $

1. Let $ B = a^{100x+10y+n} $, if $n$ is fixed then all $B$ having $y$ even will have same last two digits. $x \geq 0 $ , $ y \neq 0$
2. Let $ B = a^{100x+10y+n} $, if $n$ is fixed then all $B$ having $y$ odd will have same last two digits. $x \geq 0 $ , $ y \neq 0$
3. Let $ B = a^{100x+10y+n} $, if $n$ and $y$ are fixed then all $B$ will have same last three digits. $x \geq 0 $ , , $ y \neq 0$

And, in General

4. Let $ B = a^{m} $, if last $r$ digits of $m$ are fixed then, all $B$ will have same $(r+1)$ digits. $ r \geq 2$

For example:

• $ 13^2 $ and, $ 13^{42} $ have same last two digits i.e., 69 (See observation '1')
• $ 143^{999} $ and, $ 143^{999999} $ have same last four digits i.e., 3007 (See observation '4')

Answer 1

See Euler's theorem with $\varphi(10^m) = 4 \times 10^{m-1}$.