# Last Digits Curious Observation

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

1. 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')

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