What is the last digit of the number:
8^(7^(6^(5^(4^(3^(2^1))))))
[(a)^b implies 'a' raised to the power of 'b', ((a)^b)^c implies 'a' raised to the power 'bc', but a^(b^c) implies 'a' raised to the power 'b' raised to the power 'c'.]
8
The first three numbers are the key.
The number 8 when raised to succeeding powers ends in numbers with the following pattern:
8,4,2,6,8,4,2,6 . . .
or generally, when 8 is raised to the "x" power, the final digit will be based on the Remainder of (x/4):
Rem
1 8
2 4
3 2
4 6
So if we can divide the result of (7^((()))) by 4, the remainder should tell us the final digit.
If we rewrite 7 as (8 - 1) then raise it to the second power, we get:
(8 - 1)^2 = (64 - 16 + 1)
Since the first two numbers are divisible by 4, the remainder for the expression when divided by 4 is 1
(8 - 1)^3 = (512 - 128 + 8 - 64 + 16 - 1)
Since every number except the last is divisible by 4, the remainder for the expression when divided by 4 becomes 3.
The final element of the expression flips from +1 to -1 and the remainder flips from 1 to 3 when 7 is raised to even and odd numbers respectively.
For (6^((()))) the number is always even, so the remainder for (7^((())))/4 is always 1. So the final digit for (8^((()))) will be "8"
Solution
