perplexus.info :: Numbers : Ten-thousands Digit Determination

Ten-thousands Digit Determination (Posted on 2023-08-06)

Determine the ten-thousands digit of N, where:

N = 5^{5^{5^5⁵}}

*** Computer program assisted solutions are welcome, but a semi-analytic (hand calculator and p&p method) is preferred.
*** N is equal to 5^(5^5^5^5) and NOT equal to (((5^5)^5)^5)^5).

No Solution Yet Submitted by K Sengupta

Rating: 4.0000 (2 votes)

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Maybe solution

Comment 3 of 3 |

I feel like I've confused this somewhere.

The last five digits of 5^n repeat in a cycle of length 8 (once n>4).

So we need the value of 5^5^5^5 mod 8.

Powers of 5 alternate: 5^(even)=1 mod 8 and 5^(odd)=5 mod 8. 5^5^5^5 is odd.

So the last 5 digits of 5^N is the same as 5^5 = 03125.

The answer is 0.

Posted by Jer on 2023-08-07 09:48:51

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