Similarities in the digits of the powers of 2 and 5

Question

Many may have noticed that the negative powers of 5 contain the same digits as the positive powers of 2: This pattern intrigued me. I started to wonder if it exists in different number bases. I soon realized that, for a base n, such a pattern seems to exist when you pick a pair of factors of n. As an example, in base 6, the digits will be the same for powers of 2 and 3: For number bases where n is a square number, that means there is a symmetry in the digits for the powers of √n. For example, let's look at the powers of 3 in base 9:

And, finally if n has multiple pairs of factors, the pattern works for each pair individually. Let's take base 12, with pairs of factors (2,6) and (3,4): I wonder why this pattern seems to universally emerge in these scenarios - and if the pattern is real at all! Is there anyone who has an idea on this?

Answer 1

This is a consequence of the fact that $10 = 2 \times 5$, and thus:

$$5^{-n} = \frac{1}{5^n} = \frac{2^n \times 5^n \times 10^{-n}}{5^n} = 2^n \times {10}^{-n}$$

IOW, a negative power of $5$ is a positive power of $2$ scaled by the same power of $10$ (which means having the same digits, but shifting the decimal point). And vice versa.

Of course, this pattern generalizes to base 12 with $12 = 2 \times 6 = 3 \times 4$, and to any factorable (non-prime) base.

Answer 2

For base $n$, and any pair of $ab = n$, $b^{-k} =\left ({n\over a}\right)^{-k} = n^{-k} a^k$. The part $n^{-k}$ basically is just shifting period left or right in base $n$ system.