$(1$ point$)$ This question concerns primality testing. Consider the composite number $231.$

Recall Fermat's Little Theorem:

For any prime $p$ and integer $a,a^{p-1}-=1,$modp.

It happens that the converse to FLT is often but not always true. That is$,$ if $n$ is composite and $a$ is an integer,

then more often than not $a^{n-1}1$modn. We can use this as the basis of a simple primality test, called

the Fermat Test.

For $ain{Z}_{231}$ we make the following definitions.

We call $a$ a Fermat Liar for $n=231$ if $a^{n-1}-=1$modn, where $a!in(0,1,n-1).$

We call $a$ a Fermat Witness for $n=231$ if $a^{n-1}1$modn, where $a!in(0,1,n-1).$

Please answer the following.

a$)$ What is the first Fermat Liar in $Z_{231}?$

b$)$ What is the first Fermat Witness in $Z_{231}?$

c$)$ How many Fermat Liars are in $Z_{231}?$ Remember not to count $0,1,$ or $n-1.$