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Moderate

Order of a Reduced Residue

ABSALG-252NXG

Let $m\ge 1$ be an integer and let $a$ be any integer coprime to $m$ ($(a,m)=1$).

We define the order of $a$ mod $m$ to be the smallest positive integer $h$ such that $a^h\equiv 1$ mod $m$. (In some texts this is said as follows: $a$ belongs to the exponent $h$ mod $m$.)

Recall that a set $R$ of reduced residues mod $m$ consists of a set $R$ of integers coprime to $m$ that are distinct mod $m$, and such that every integer $a$ coprime to $m$ satisfies $a\equiv r$ for some $r\in R$.

Let $R$ be a set of reduced residues mod $12$.

Which of the following is TRUE?

Select ALL that apply.

A

There are exactly three elements of $R$ that have order $2$.

B

Every $r\in R$ has order $2$.

C

There are two elements of $R$ that have order $2$ and two elements of $R$ that have order $4$.

D

There is one element of $R$ of order $1$, two of order $2$, and one of order $4$.

E

There are no elements of $R$ of order $\varphi(12)$.