1 to the 4th power is 1.

2 to the 4th power is 16.

3 to the 4th power is 81

4 to the 4th power is 256

5 to the 4th power is 625

These are evenly divisible by 5 with a remainder of 1.

Observe that the remaining numbers from 6 to 9 raised to the 4th power match up with the digits in the one position of the first four I've already raised to the 4th power. The rest, 10 and above, are trivial. (PS I haven't been to college in over 30 years myself)