$1.$ Suppose you have a $32-$bit MIPS processor. For each of the two following cache configurations, how many bits are used for the tag field, the set field $($also called the index$),$ and the byte offset field of the address?

$($a$)$ A direct mapped cache with capacity $8192$ bytes and block size $32$ bytes.

$($b$)$ A $8-$way set associate cache with capacity $2048$ bytes and block size $16$ bytes.