Consider a set-associative cache of size 2KB ($1\text{KB} = 2^{10}$ bytes) with cache block size of 64 bytes. Assume that the cache is byte-addressable and a 32-bit address is used for accessing the cache. If the width of the tag field is 22 bits, the associativity of the cache is ________.

Given parameters:

• Cache size $= 2\text{ KB} = 2 \times 2^{10}\text{ bytes} = 2048\text{ bytes}$

• Block size $= 64\text{ bytes} = 2^6\text{ bytes}$

• Total address bits $= 32\text{ bits}$

• Tag bits $= 22\text{ bits}$

1. Block Offset Bits:

   $$\text{Block offset} = \log_2(64) = 6\text{ bits}$$

2. Set Index Bits:

   $$\text{Set index bits} = \text{Address bits} - (\text{Tag bits} + \text{Block offset bits})$$

   $$\text{Set index bits} = 32 - (22 + 6) = 32 - 28 = 4\text{ bits}$$

3. Number of Sets:

   $$\text{Number of sets} = 2^4 = 16$$

4. Total Number of Cache Blocks:

   $$\text{Total blocks} = \frac{\text{Cache size}}{\text{Block size}} = \frac{2048}{64} = 32$$

5. Associativity ($k$-way):

   $$k = \frac{\text{Total blocks}}{\text{Number of sets}} = \frac{32}{16} = 2$$

Thus, the associativity of the cache is 2.