Or another way to look at it:
0th power of 12357 ends in 1.
1st power ends in 7. 2nd power ends in 9. 3rd power ends in 3. 4th power ends in 1
5th power ends in 7 6th power ends in 9 7th power ends in 3. 8th power ends in 1
9 ... 7 10 ... 9 11 ... 3 12 ... 1, etc.
There are only four digits that occur in the powers and these repeat in the sequence 1, 7, 9, 3, 1, ...
Divide the power (655 in this case) by four and see what the remainder is. It is three. So the 655th
power ends in the same digit that the 3rd power does, namely 3.
But this is a bit confusing since we get 3 for the remainder and for the last digit.
Try raising to the 656 power. Then dividing 656 by 4 we get a remainder of zero. So the 656
power ends in 1 same as the 0th power.
For the 654th power dividing 654 by 4 has remainder 2. So the 654th power ends in 9 the same
as the second power.
For the 653rd power dividing 653 by 4 has remainder 1 so the units digit of the 653rd power is 7 the
same as the 1st, 5th, 9th, ... powers.
Be very blesses!