This question is about digital sequences. If it helps, please use the claims of the questions prior to prove later ones, even if you do not know the answer to the previous question. Please also help to proove all claims and explain all calculation.

$1.$ Consider the LFSR of order $n$ over the binary alphabet, $F_2,$ with the following feedback function:

$f(x_1,x_2,$dots,$x_n)=x_1$

i$)$ Prove that this LFSR is non$-$singular.

ii$)$ Prove that all of its output sequences have a period $n.$

iii$)$ Let $d$ be any positive integer, $1dn,$ that divides $n,$ namely, $d|n.$ Show that this LFSR has an output sequence with minimal period $d.$

iv$)$ Now let $d$ be any positive integer, $1dn,$ that does not divide $n,$ namely, $dn.$ Prove that this LFSR does not have any output sequence with minimal period $d.$

v$)$ Let $p$ be a prime number. Prove that $p|2^p-2.$

Hint: Look at the state diagram of the LFSR in this question with $n=p.$