$3.$ Work out the following problems from first principles without converting to binary and counting the bits. Then compare with the formulae presented for encoding a character string into an integer, and vice$-$versa, in the specified radix. $($Hint: Consider the next$-$lower and next$-$higher power of two for each integer.$)$

a$)$ How many bits are exactly required to encode the following integers? $($The number shown as an integer$$s subscript refers to the radix of that integer.$)$

b$)$ Exactly how many bytes are required to represent the numbers in $($a$)$ above?

$i.{8191}_{10}$

$ii.{8192}_{10}$

iii. ${65535}_{10}$

$iv.{65536}_{10}$

$v.{65535}_{16}$

$vi.{65536}_{16}$

vii. $34F{7}_{16}$

viii. ${13559}_{10}$