Problem $6(20$ points$).$ Prove that the operation $*($multiplication$)$ on $Z$ satisfies the following

properties $(4$ points each$)$:

$($a$)$ Distributivity: $m*(n+p)=m*n+m*p$ for all $m,n,$pinZ.

$($b$)$ Associativity: $(m*n)*p=m*(n*p)$ for all $m,n,$pinZ.

$($c$)$ Commutativity: $m*n=n*m$ for all $m,$ninZ.

$($d$)$ Multiplicative unit $($i$.$e$.$ one$)$: the element $1$ satisfies the property that $m*1=m$ for all

minZ. Moreover, any element with this property has to be $1$ itself. The element $1$ is

called the multiplicative unit.

$($e$)$ Cancelation: $m*k=n*k$ implies $m=n$ for all $m,n,$kinZ such that $k0.$