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%(NZM, Problem ) 

\ctln{\bf Math 445 Homework 4}

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\ctln{Due Friday, October 3}

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\item{13.} Show that if $n|m$, and $(10,m)=1$, then the 
period of the decimal expansion of $1/n$ divides the
period of the decimal expansion of $1/m$ .

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\item{14.} Show that for every $n\geq 2$, $\text{ord}_{3^n}(10)=3^{n-2}$ .

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\item{} (Hint: induction! This is not entirely unlike what we did for $7^n$....) 

\item{} [N.B.: Consequently, the period of the decimal expansion of $1/3^n$ is $3^{n-2}$ .]

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\item{15.} Show that if $(3,n)=1$ ( and $(10,n)=1$ ), 
then $\text{ord}_{n}(10)=\text{ord}_{3n}(10)=\text{ord}_{9n}(10)$ .

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\item{16.} Find the primitive roots of 1 mod 31. (I.e., find all $a$, $1\leq a\leq 31$, with
$\text{ord}_{31}(a)=30$. 

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\item{} (Hint: find one; then use one of our results to quickly find the others.)

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