A computer uses $46-bit$ virtual address, $32-bit$ physical address, and a three–level paged page table organization. The page table base register stores the base address of the first-level table ($T1$), which occupies exactly one page. Each entry of $T1$ stores the base address of a page of the second-level table ($T2$). Each entry of $T2$ stores the base address of a page of the third-level table ($T3$). Each entry of $T3$ stores a page table entry ($PTE$). The $PTE$ is $32$ bits in size. The processor used in the computer has a $1$ MB $16$ way set associative virtually indexed physically tagged cache. The cache block size is $64$ bytes.
What is the minimum number of page colours needed to guarantee that no two synonyms map to different sets in the processor cache of this computer?
- $2$
- $4$
- $8$
- $16$