A computer system with a word length of 32 bits has a 16 MB byte-addressable main memory and a 64 KB, 4-way set associative cache memory with a block size of 256 bytes. Consider the following four physical addresses represented in hexadecimal notation. A1 = 0x42C8A4, A2 = 0x546888, A3 = 0x6A289C, A4 = 0x5E4880 Which one of the following is TRUE?

### 1. Cache Geometry & Address Breakdown:

• Main Memory Size $= 16\text{ MB} = 2^{24}\text{ bytes} \implies$ Physical Address is $24\text{ bits}$.

• Block Size $= 256\text{ bytes} = 2^8\text{ bytes} \implies \text{Block Offset} = 8\text{ bits}$.

• Cache Size $= 64\text{ KB} = 2^{16}\text{ bytes}$.

• Total Number of Cache Lines $= \frac{2^{16}}{2^8} = 256$.

• Since it is 4-way set associative, the number of sets is:

  $$\text{Number of Sets} = \frac{256}{4} = 64 = 2^6 \implies \text{Set Index} = 6\text{ bits}$$

• Tag Bits $= 24 - (6 + 8) = 10\text{ bits}$.

Address division (24 bits):

### 2. Extracting the Set Index for Each Address: The lower 8 bits (the last 2 hex digits) correspond to the block offset. The next 6 bits (bits 8 to 13) correspond to the set index, which is given by $((\text{Address} \gg 8) \ \& \ 0\text{x}3\text{F})$:

• A1 = 0x42C8A4:

  • $\text{Address} \gg 8 = \text{0x42C8}$

  • Lowest byte is $\text{0xC8} = 11001000_2$

  • Lowest 6 bits $= 001000_2 = 8_{10}$

  • $\text{Set}(A1) = 8$

• A2 = 0x546888:

  • $\text{Address} \gg 8 = \text{0x5468}$

  • Lowest byte is $\text{0x68} = 01101000_2$

  • Lowest 6 bits $= 101000_2 = 40_{10}$

  • $\text{Set}(A2) = 40$

• A3 = 0x6A289C:

  • $\text{Address} \gg 8 = \text{0x6A28}$

  • Lowest byte is $\text{0x28} = 00101000_2$

  • Lowest 6 bits $= 101000_2 = 40_{10}$

  • $\text{Set}(A3) = 40$

• A4 = 0x5E4880:

  • $\text{Address} \gg 8 = \text{0x5E48}$

  • Lowest byte is $\text{0x48} = 01001000_2$

  • Lowest 6 bits $= 001000_2 = 8_{10}$

  • $\text{Set}(A4) = 8$

### 3. Evaluating the Options:

• A: A1 and A4 are mapped to the same cache set ($8$). (False)

• B: A2 and A3 are mapped to the same cache set ($40$). (True)

• C: A3 ($40$) and A4 ($8$) are mapped to different cache sets. (False)

• D: A1 ($8$) and A3 ($40$) are mapped to different cache sets. (False)

Thus, statement B is TRUE.