Let $n $\backslash$in $\backslash$mathbb$\{$Z$\}$$ be a positive integer and $n $=$ p$\_1^\{$a$\_1\}\backslash$cdots p$\_$r$^\{$a$\_$r$\}$$ its prime factorization. By the classification theorem for finite abelian groups

$\backslash [$

$\backslash$frac$\{\backslash$mathbb$\{$Z$\}\}\{($n$)\}\backslash$cong $\backslash$frac$\{\backslash$mathbb$\{$Z$\}\}\{($p$\_1^\{$a$\_1\})\}\backslash$times $\backslash$cdots $\backslash$times $\backslash$frac$\{\backslash$mathbb$\{$Z$\}\}\{($p$\_$r$^\{$a$\_$r$\})\}$

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as $\backslash$textit$\{$abelian groups$\}.$

$\backslash$begin$\{$itemize$\}$

$\backslash$item Use the CRT to prove that this is in fact a $\backslash$textit$\{$ring$\}$ isomorphism.

$\backslash$item Prove that

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$\backslash$left$(\backslash$frac$\{\backslash$mathbb$\{$Z$\}\}\{($n$)\}\backslash$right$)^\{*\}\backslash$cong $\backslash$left$(\backslash$frac$\{\backslash$mathbb$\{$Z$\}\}\{($p$\_1^\{$a$\_1\})\}\backslash$right$)^\{*\}\backslash$times $\backslash$cdots $\backslash$times $\backslash$left$(\backslash$frac$\{\backslash$mathbb$\{$Z$\}\}\{($p$\_$r$^\{$a$\_$r$\})\}\backslash$right$)^\{*\}$

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$($recall that $$(\backslash$mathbb$\{$Z$\}/$n$\backslash$mathbb$\{$Z$\})^\{*\}$$ denotes the group of units of $$\backslash$mathbb$\{$Z$\}/$n$\backslash$mathbb$\{$Z$\}$$$.)$

$\backslash$item Recall that Euler$$s $$\backslash$phi$$-$function $$\backslash$phi$($n$)$$ denotes the number of primes $