Xinmeng Z.

asked • 01/18/19

What are the last digits of 3^3^3^3

If Xn=3^3^3.(n times)..^3 with n bigger than 10, show that their 10 last digits are the same. I think we should use Euler Fermat's theorem, and calculate with modulo(10^10) and somehow prove it by induction. phi(10¹⁰)=phi(2¹⁰)*phi(5¹⁰)=4*10⁹


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Mark M. answered • 01/18/19

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If I read your expression correctly, it is equal to 327.

The powers of three are:

3, 9, 27, 81, 243, 729, ...

If only the last digit is considered.

27 / 4 = 6.75

So the last digit is 7.

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Xinmeng Z.

the three last digits. I found 187，but I＇m Not sure. The next question is to prove that all the 3^(3^3^...3) have the same last 10 digits, if there are more that ten 3 at the top.
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01/18/19

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