[Math] Finding the last two digits of a number

elementary-number-theory

I came across this problem.
I have to find the last two digits ,and if I can , the last three digits of this number (which I believe is Infinity.)

$$n={2017}^{({2018)}^{{(2019)}^{{(•)}^{{(•)}^{(•)}}}}}$$
I started by computing $n \pmod {10} $ and which I think is $$7^{2k} \equiv 1 {\pmod {10}}$$ where $k$ is even …

Next , I tried calculating $n{\pmod {100}}$ and I believe the answer is either $21,41,61,81$ but I don't know for sure..It was lot of trial and error…

So Could you please check my answer and ,if wrong, provide me a hint in the right direction?

Best Answer

Let's assume the power tower is finite but includes at least the $2020$ term and possibly many more

We can say:

  • $2020^n$ is even, i.e. of the form $2m$ when $n\ge 1$
  • $2019^{2m}$ is of the form $4l+1$, as are all squares of odd numbers
  • $2018^{4l+1}$ is of the form $100k+68$ when $l \ge 1$
  • $2017^{100k+68}$ is of the form $1000j+241$

suggesting the final three digits are $241$

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