Exercise $3\mathrm{.}2$: Turing numbers

Show that single$-$tape Turing machines can each be given a number from the list $1,2,3,$dots in such a way

that the number uniquely specifies the corresponding machine. We call this number the Turing number

of the corresponding Turing machine. $($Hint: Every positive integer has a unique prime factorization

$p_1^{a_1}{p}_2^{a_2}dots{p}_k^{a_k},$ where $p_i$ are distinct prime numbers, and $a_1,dots{a}_k$ are non$-$negative integers.$)$

Exercise $3\mathrm{.}3$: Turing machine to reverse a bit string

Describe a Turing machine which takes a binary number $x$ as input, and outputs the bits of $x$ in reverse

order. $($Hint: In this exercise and the next it may help to use a multi$-$tape Turing machine and$/$or symbols

other than $,0,1$ and the blank.$)$