Use the given equations in a complete proof of each theorem. Your proof should be expressed in complete English sentences.

$($a$)$

Theorem: If a$,$ b$,$ and c are integers such that a$3|$b and b$2|$c$,$ then a$6|$c$.$

b$=$k$$a$3$

c$=$j$$b$2$

c$=$j$$b$2=$j$($k$$a$3)2=($j$$k$2)$a$6$

$($b$)$

Theorem: If m and n are integers such that m$|$n$,$ then m$|(5$n$32$n$2+3$n$).$

n$=$k$$m

$5$n$32$n$2+3$n$=5($k$$m$)32($k$$m$)2+3($k$$m$)=5$k$3$m$32$k$2$m$2+3$k$$m$=(5$k$3$m$22$k$2$m$+3$k$)$m

$($c$)$

Theorem: If n is an odd integer, then $4$ divides n$21.$

n$=2$k$+1$

n$21=(2$k$+1)21=(4$k$2+4$k$+1)1=4($k$2+$k$)$

$($d$)$

Theorem:The sum of the squares of any two consecutive integers is odd.