In Part 6 of this series, I mentioned the following fact concerning the decimal representation of : if neither nor is a factor of , then the repeating block in the decimal representation of has a length that must be a factor of . This function is the Euler toitent function or the number of integers less than that are relatively prime with .
In this post, I’d like to provide a justification for this theorem.
As discussed earlier, is the least integer so that is a factor of . In the language of congruence, is the least integer so that
In other words, let be the multiplicative group of numbers less than that are relatively prime with . By assumption . Then is the order of in , 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 is the number of integers less than that are relatively prime with , or .
In other words, using these ideas from group theory, we can prove that .
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.