Let m$,$ n be integers such that gcd$($m$,$ n$)=1.$

$($i$)$ Show that there is a group isomorphism Z

$\backslash$times

nm

$=$ Z

$\backslash$times

n $\backslash$times Z

$\backslash$times

m$.$

$($ii$)$ Recall that $\backslash$phi $($n$)=|$Z

$\backslash$times

n

$|$ is the order of the units modulo n$.$ Show that $\backslash$phi $($nm$)=$

$\backslash$phi $($n$)\backslash$phi $($m$).$

Hint. An additive version of this is explained in Proposition $3\mathrm{.}4\mathrm{.}5.$ Whatever isomorphism you

propose to define, make sure that it is well$-$defined!