You should indicate your exponents more carefully.
It was almost impossible to read your post . . .
They say, 100 is a multiple of 4, that's why the units digit of is the same as that of .
However, 100 is also a multiple of 5. So why did they pick 4?
If is an integer with units digit 3, what is the units digit of ?
Since only the units digit of matters, you can start by exploring what happens when
Raise 3 to powers and observe the units digits of the results.
Notice that the units digit of is again 3, so from here on the sequence
will repeat itself and continue to cycle through the four values 3, 9, 7, 1.
Since 100 is a multiple of 4, the units digit of is the same as that of
I think you answered your own question . . .
Since ends in 1, we can write: .
Raise both sides to the 25th power: .
. . and we have: .