There is a power of 3 that ends in 01.
There is a power of 3 that ends in 001.
There is a power of 3 that ends in 0001.
There is a power of 3 that ends in however many 0's you like and then a 1.
:O
For the intrepid reader: Why?
Take the first 101 powers of 3 (31, 32, ... 3101). There are only 100 possibilities for the last 2 digits (00 to 99), so some 2, let's say the nth and mth powers, have the same last 2 digits! But that means that 3m - 3n, or 3n (3m-n - 1), will have 00 in the last 2 digits, so it will be divisible by 100. Well, 3n certainly doesn't have any factors in common with 100, so that means that 3m-n - 1 is divisible by 100. In other words, 3m-n will have a remainder of 1 when we divide by 100. But that means that it ends in 01! YAY!!! :)
The cool part is that you can do exactly the same thing for 1,000, 10,000, etc., so you can have as many 0's as you like!
You can prove some crazy things with powers and modular arithmetic.
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