How to find the last digit of exponentiation of a series of numbers?

Question

Mathematics

What is the best way to find the last digit of the expression x1 ^ (x2 ^ (x3 ^ (x4 ^ ( ... ^xn)))), given that x values ​​can be very large (greater than six digits)?
In theory, it is enough for us to know only the last digit of x1 and the last two digits of the outer bracket, but what is the algorithm for reducing the remaining terms?

Answer 1

Solution of the problem in C#:

public static int LastDigit(int[] array) {
      if (array.Length == 0) return 1;
      array[0] %= 10;
      if (array.Length == 1 || array[0] == 1) return array[0];
      
      
      for (int i = 1; i < array.Length && i < 4; i++) {
        if (array[i] == 0) {
          int j = 0;
          while(i + j + 1 < array.Length && array[i + j + 1] == 0) { j++; }
          Array.Resize(ref array, i);
          if (j % 2 == 0) { array[i - 1] = 1; }
          if (i == 1) return array[0];
          break;
        }
      }
      
      if (array[0] == 0 || array[0] == 5 || array[0] == 6) return array[0];
      if (array[0] == 4) { return (array[1] % 2 == 0) ? 6 : 4; }
      if (array[0] == 9) { return (array[1] % 2 == 0) ? 1 : 9; }


      if (array.Length > 2) array[1] = (int)BigInteger.ModPow(array[1], array[2], 100);
      if (array[1] == 0) array[1] = 4;
      return (int)BigInteger.ModPow(array[0], array[1], 10);
}

Answer 2

You need the result of exponentiation in the ring of residues modulo 10, you can start here:
https://ru.wikipedia.org/wiki/Raising_to_power...
You can also see the part about number theory here:
https://stepik. org/course/4603