# Thoughts on 1/7 and other rational numbers (Part 8)

In Part 6 of this series, I mentioned the following fact concerning the decimal representation of $\displaystyle \frac{a}{b}$: if neither $2$ nor $5$ is a factor of $b$, then the repeating block in the decimal representation of $\displaystyle \frac{a}{b}$ has a length $k$ that must be a factor of $\phi(b)$. This function is the Euler toitent function or the number of integers less than $b$ that are relatively prime with $b$.

In this post, I’d like to provide a justification for this theorem.

As discussed earlier, $k$ is the least integer so that $b$ is a factor of $10^k - 1$. In the language of congruence, $k$ is the least integer so that

$10^k \equiv 1 (\mod b)$

In other words, let $G_b$ be the multiplicative group of numbers less than $b$ that are relatively prime with $b$. By assumption $10 \in G_b$. Then $k$ is the order of $10$ in $G_b$, and there’s a theorem that states that the order of an element of a group must be a factor of the order of the group, or the number of elements in the group. In our case, the order of $G_b$ is the number of integers less than $b$ that are relatively prime with $b$, or $\phi(b)$.

In other words, using these ideas from group theory, we can prove that $k \mid \phi(b)$.

Naturally, we don’t expect middle school students seeing long division for the first time to appreciate this property of decimal representations. Still, my main purpose in writing this post was to give a concrete example of how ideas from higher-level mathematics — like group theory — actually can shed insight into ideas that are first seen in school — even middle school. In other words, there’s a reason why UNT (and other universities) requires that college students who want to earn mathematics teaching certification with their degrees must have a major in mathematics.

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