A Pattern Among Powers

    In Maths class, a while ago, my teacher made a table,
showing the last digits of exponents, and pointed out a
pattern. For example, take the number 7. If we consider
the last digit of increasing powers of 7 (7¹, 7², 7³...),
then we see that they follow a regular pattern. (7 - 9 - 3 - 1)

7¹ = 7
7² = 49
7³ = 343
7⁴ = 2401

7⁵ = 16807
7⁶ = 117649
7⁷ = 823543
7⁸ = 5764801


Another interesting observation attained from this can
be that every fourth power of 7 (7⁴, 7⁸, 7¹²...), ends in -01.
This is because every fourth power is also a power of 7⁴,
which is 2401.

But this was not the most fascinating thing about the table.
Since the table was for all 10 digits (0-9), one column was
for the loop lengths, or the number of exponents until a digit
recurred. For 7, the loop length would be 4. For numbers like
5, 6, 1 and 0, the loop length is 1.
One of my classmates pointed out that the loop lengths of the
recurring digits also recurred.

Ending Digit            Loop Length

0                                       1
1                                       1
2                                       4
3                                       4
4                                       2
5                                       1
6                                       1
7                                       4
8                                       4
9                                       2

As we can see, the length of the loops goes in a loop of
1-1-4-4-2. Interesting. Since this post takes up quite a
bit of space as is, I'll include my investigation of this
phenomenon in the next post. I'd like to see whether this
holds true for other bases, and hopefully find an
explanation.