Too Many 5s

Number Theory Level 4

Find the $5$ th last digit from the right of the decimal representation of $5^{5^{5^{5^5}}}$ .

1 solution

Charlz Charlizard
Dec 30, 2019

So to solve this problem I used the following logic.

I found the pattern for the 5th digit in the power of 5's.
So after 5th power the pattern cycle goes like this: (1,1), (2,7), (3,9), (4,5), (5,6), (6,2), (7,4), (8,0). The cycle repeats after 8th power after 5th.
So now we will remove the first 5 powers of 5 as the cycle starts after 5th power of as mentioned before.
Now we will find the remainder of the given powers of 5, subtracting 5, when divided by 8.
So it will be (5 ^ (5 ^ 5 ^ 5)-5) % 8, which is similar as (5 ^ (5 ^ 5 ^ 5)) % 8 - 5 % 8 (%= modulo operator)
Now 5 ^ (5 ^ 5 ^ 5) % 8 = (-3) ^ (5 ^ 5 ^ 5) % 8 = (-1) * 3 ^ (5 ^ 5 ^ 5) % 8 = (-3) * 9^((5 ^ 5 ^ 5)-1)/2) % 8 = (-3) * 1^ ((5 ^ 5 ^ 5-1)/2) % 8 = -3
So remainder will be 8 - 3 = 5
Now the actual question (5^(5^5^5)) % 8 - 5 % 8 = 5 - 5 = 0.
And so the 5th digit of the given powers of 5 will be 0

DO NOT MIND THE FORMATTING :-)