explain me this question and solve it side by side "Consider a direct mapped cache of size 16 KB with block size 256 bytes. The size of main memory is 128 KB. Find- Number of bits in tag Tag directory size" ChatGPT Sure, I can help you with that question! In a direct-mapped cache, each memory block can be mapped to only one location in the cache. The cache is divided into a number of fixed-size blocks, and each block can store one memory block. In this case, the cache size is 16 KB and the block size is 256 bytes. To calculate the number of blocks in the cache, we can use the following formula: Number of blocks = cache size / block size Number of blocks = 16 KB / 256 bytes = 64 blocks Since this is a direct-mapped cache, each memory block can be mapped to only one block in the cache. Therefore, we need to determine how many bits are required to represent the block number in the cache. Since we have 64 blocks in the cache, we can represent each block using 6 bits (2^6 = 64). Therefore, the tag bits will be the remaining bits after we use the 6 bits for the block number. Tag bits = total address bits - block bits To calculate the total address bits, we can use the size of main memory, which is 128 KB. Since the block size is 256 bytes, we need to represent each memory block using 8 bits. Total address bits = log2(main memory size / block size) Total address bits = log2(128 KB / 256 bytes) Total address bits = log2(512) Total address bits = 9 bits Therefore, the tag bits will be: Tag bits = 9 bits - 6 bits = 3 bits Now we can calculate the tag directory size. In a direct-mapped cache, the tag directory stores the tag bits for each cache block. Since we have 64 blocks in the cache, we need to store 64 tag bits. Each tag bit takes up 1 bit of storage in the tag directory. Therefore, the tag directory size will be: Tag directory size = number of tag bits * number of blocks Tag directory size = 3 bits * 64 blocks = 192 bits So the answer to the question is: Number of bits in tag = 3 bits Tag directory size = 192 bits User Consider a direct mapped cache of size 512 KB with block size 1 KB. There are 7 bits in the tag. Find- Size of main memory Tag directory size ChatGPT Sure, I can help you with that! In a direct-mapped cache, each memory block can be mapped to only one location in the cache. The cache is divided into a number of fixed-size blocks, and each block can store one memory block. In this case, the cache size is 512 KB and the block size is 1 KB. To calculate the number of blocks in the cache, we can use the following formula: Number of blocks = cache size / block size Number of blocks = 512 KB / 1 KB = 512 blocks Since this is a direct-mapped cache, each memory block can be mapped to only one block in the cache. Therefore, we need to determine how many bits are required to represent the block number in the cache. Since we have 512 blocks in the cache, we can represent each block using 9 bits (2^9 = 512). Therefore, the tag bits will be the remaining bits after we use the 9 bits for the block number. Tag bits = total address bits - block bits In this case, we are given that there are 7 bits in the tag. Therefore, the total address bits can be calculated as: Total address bits = tag bits + block bits Total address bits = 7 bits + 9 bits = 16 bits Now we can calculate the size of main memory. The size of main memory is the total number of memory blocks multiplied by the block size. Size of main memory = number of memory blocks * block size Size of main memory = 2^(total address bits) * block size Size of main memory = 2^16 * 1 KB Size of main memory = 65536 KB = 64 MB Therefore, the size of main memory is 64 MB. We can also calculate the tag directory size. In a direct-mapped cache, the tag directory stores the tag bits for each cache block. Since we have 512 blocks in the cache, we need to store 512 tag bits. Each tag bit takes up 1 bit of storage in the tag directory. Therefore, the tag directory size will be: Tag directory size = number of tag bits * number of blocks Tag directory size = 7 bits * 512 blocks = 3584 bits So the answer to the question is: Size of main memory = 64 MB Tag directory size = 3584 bits User Consider a direct mapped cache with block size 4 KB. The size of main memory is 16 GB and there are 10 bits in the tag. Find- Size of cache memory Tag directory size 3. No. of physical address bits 4. No. of bits in block offset ChatGPT Sure, I can help you with that! In a direct-mapped cache, each memory block can be mapped to only one location in the cache. The cache is divided into a number of fixed-size blocks, and each block can store one memory block. In this case, the block size is 4 KB. To calculate the number of blocks in the cache, we need to know the size of the cache. However, we are not given the size of the cache directly. We can find the size of the cache by using the tag bits and the block offset bits. Tag bits = 10 bits (given) Block offset bits = log2(block size) = log2(4 KB) = 12 bits The remaining bits will be used for the block number, which will be used to index into the cache. Therefore, the number of blocks in the cache can be calculated as: Number of blocks = 2^(number of block number bits) Number of blocks = 2^(total number of address bits - tag bits - block offset bits) Number of blocks = 2^(log2(main memory size) - tag bits - block offset bits) Number of blocks = 2^(log2(16 GB) - 10 - 12) Number of blocks = 2^(34 - 10 - 12) Number of blocks = 2^12 = 4096 Therefore, the size of the cache memory is: Size of cache memory = number of blocks * block size Size of cache memory = 4096 * 4 KB Size of cache memory = 16 MB The tag directory size is the total number of tag bits required to store the tags for all the blocks in the cache. In this case, we have 4096 blocks in the cache, and each block requires 10 tag bits. Therefore, the tag directory size will be: Tag directory size = number of tag bits * number of blocks Tag directory size = 10 bits * 4096 blocks Tag directory size = 40,960 bits The number of physical address bits is the total number of bits required to represent a physical address. In this case, the physical address consists of the tag bits, the block number bits, and the block offset bits. Number of physical address bits = tag bits + number of block number bits + block offset bits Number of physical address bits = 10 + log2(number of blocks) + 12 Number of physical address bits = 10 + log2(4096) + 12 Number of physical address bits = 22 Therefore, the number of physical address bits is 22. The number of bits in block offset is the number of bits required to represent the offset within a block. In this case, we have a block size of 4 KB, which requires 12 bits to represent the block offset. Therefore, the number of bits in block offset is 12 bits. So the answers to the questions are: Size of cache memory = 16 MB Tag directory size = 40,960 bits No. of physical address bits = 22 No. of bits in block offset = 12