find the unit digit of 82^83^84^85..can u explain its solution..i.e how it is simplified in the form of 4k+1

Hi Abhinav,

In order to simplify $$82^{83^{84^{85}}}$$, you have to first simplify the indices and then come to the base.
The unit digit of $$82^x$$ depends on whether x is of the form 4k, 4k+1, 4k+2 or 4k+3.
$$84^{85}$$ is a multiple of 84, which is of the form 4k. Hence, $$84^{85}$$ is of the form 4k.
$$83^{4k}$$ always ends with unit digit 1 and ten's digit as an even number. Hence $$83^{4k}$$ is of the form 4k+1.
$$82^{4k+1}$$ ends with 2.

Hence, the unit digit of $$82^{83^{84^{85}}}$$ is 2.