$1.$ In a computer system, the virtual address space requires $36$ bits, and the physical address space requires $31$ bits. The page size is $128$ MB$.$ Additionally, the system uses a TLB $($Translation Lookaside Buffer$)$ with $64$ entries. Answer the following questions to analyze the virtual memory system:

a$.$ Calculate the number of page table entries.

b$.$ Given the system uses one I$/$V $($invalid$/$valid$)$ bit, one dirty bit and one reference bit for each page table entry, calculate the size $($in bytes$)$ of the page table.

c$.$ Calculate the size $($in bytes$)$ of the TLB$.$

d$.$ Assuming a uniform distribution of the TLB entries over the virtual address space, calculate the TLB hit ratio.

e$.$ If we used an inverted page table where maximum process ID is $15000,$ what would be the size of that page table