Problem $4$

Suppose that you want to encrypt a message using an affine cipher. You let $A=0,B=1,$dots.,$A=25,$ but you also include $?=26,$;$=27,Q^{\text{'}}=28,4=29.$ Therefore, you use $xa*x+b$ mad $30$ for your encryption function, for some integers a and $b.$

$($a$)$ Show that there are exactly eight possible choices for the integer a $($that is$,$ there are only eight choices of a $($with $a=10,b=00$