Problem $1[30$ pts$]$: A memory system has a $32$ KB byte$-$addressable main memory and a $1$ KB cache where each block contains $16$ bytes. Determine the format of the address in each of the cache organizations. In each case list the address fields and the width of each field.

a$)$ Direct$-$mapped cache

b$)$ Fully$-$associative cache

c$)4-$way set associative cache