SOLUTION: Find the unit digit of 64^401

The units digit of numbers 64, 64^2, 64^3, 64^4, 64^5  and so on form the cyclic sequence 4, 6, 4, 6, 4 . . . . and so on.


The length of the cycle is 2; each even-indexed number in this cycle is 6; each odd-indexed number in this cycle is 4.


So, the answer to the problem's question is 4  (since the degree index 401 is an odd number).

The unit digit of an integer is the digit it ends in.

Odd powers of positive integers ending in 4 end in 4.
Even powers of positive integers ending in 4 end in 6.

64 is an integer ending in 4 and the 401th power is an odd power,
so 64^401 ends in 4.

Edwin