How and why can it be true that some decimals that sometimes go on forever in base 10 such as $1\over 3$ change to good numbers in base 12?

Question

I've noticed on a math channel on YouTube that some of these decimals from fractions in the decimal system such as $1\over 3$, which equals $0.\overline3$, which goes on forever, doesn't show off this thing in the dozenal, or duodecimal, system: $4\over 12$ is the same as $1\over 3$ and it converts to $4\over 10$, or $4$ over do, which gets us $0.4$ if we look at it that way. Also, $1\over 4$ kindly equals $0.25$, which obviously terminates. Also, $3\over 12$ is the same as $1\over 4$, which converts to $3\over 10$ or $3$ over do, which gets us $0.3$ if we look at it that way. This can also happen for fractions $1\over 2$, $1\over 6$, and even $1\over 12$. This can also happen with base-$10$ fractions with some denominators divisible by $12$. How and why is this true?

Answer 1

Fractions terminate if all the prime factors of the denominator (once the fraction is in lowest terms) are prime factors of the base. In base $10$, fractions terminate if the denominator has no prime factors besides $2$ and $5$, so fractions with denominators like $2,5,16,125,200,$ etc will terminate. In base $12$, fractions terminate if the denominator has no prime factors other than $2$ and $3$. $\frac 19=0.14_{12}$ terminates nicely, but $\frac 15=0.\overline{2497}_{12}$ does not. If you go to base $30$ then you can afford factors $2,3$ and $5$ and have your fractions terminate. Unfortunately, your addition and multiplication tables get much larger.