Determine the tenthousands digit of
N, where:
N = 5^{5555}
*** Computer program assisted solutions are welcome, but a semianalytic (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).
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 20230807 09:48:51 