Power cycles

I will make a series of posts detailing Power cycles,Divisibility Tests,Remainders
using Basic Remainder Theorem,Fermat's Theorem,Chinese Remainder theorem ,
Constant remainders in an A P and successive remainders.

Power cycles is an easy and fairly well known concept.
However for the benefit of those who donot know what it means
I will provide some explanation .

Power Cycles can be used to find the last digit of a number raised to some
power. Its limited in it's use as we can only get the last digit.


Now the idea is 1 raised to any power will end in 1.
So the power cycle for 1 is 1 with a frequency of 1.

For 2,it can be 2, 4, 8 and then 6 .After 6 we get 2 and the cycle repeats.
As in 2 ^ 1 = 2 , 2 ^ 2 =4,2^3 =8 ,2 ^ 4 =16, 2^ 5 =32 and so on.

Similarly for 3, It is 3,9,7 and 1

For 4, it is 4 and 6

For 5 it is 5

For 6 it is 6

And for 7 it is 7,9,3 and 1

For 8 it is 8,4,2,6

For 9 it is 9 and 1

For 0 it is 0.

So 2,3,7 and 8 have power cycles with frequency 4.
0,1,5,6 have power cycles with frequency 1.
4 and 9 have a frequency 2.

Now if we have to find the last digit of 2 ^ 73,
We need to express 73 as 4k+1/4k+2/4k+3/4k (4 because the
Frequency of 2’s power cycle is 4).
Here in this case it is in the format 4k+1 , so the last digit will be 2.
If it is of the form 4k+2, the last digit will be 4 and if 4k+3 then 8 and so on.
Similarly we can use it for all other numbers.

The post is getting bigger and bigger,
Will elaborate on other concepts in a separate post.