An integer $n>1$ , whose positive divisors are $1=d_1<d_2< \cdots <d_k=n$, is called $\textit{southern}$ if all the numbers $d_2-d_1, d_3- d_2 , \cdots, d_k-d_{k-1}$ are divisors of $n$.

a) Find a positive integer that is $\textit{not southern}$ and has exactly $2022$ positive divisors that are $\textit{southern}$.

b) Show that there are infinitely many positive integers that are $\textit{not southern}$ and have exactly $2022$ positive divisors that are $\textit{southern}$.
Let's break down the solution into detailed steps to ensure clarity and completeness.

### Part (a)
1. **Identify a number with exactly 2022 divisors:**
   Consider the number \( n = 2^{2022}p \), where \( p \) is a prime number greater than \( 2^{2022} + 2^{4044} \). The number \( n \) has exactly \( 2022 \) positive divisors because the divisors are of the form \( 2^k \) and \( 2^k p \) for \( k = 0, 1, 2, \ldots, 2021 \).

2. **List the divisors:**
   The divisors of \( n \) are:
   \[
   1, 2, 4, \ldots, 2^{2022}, p, 2p, 4p, \ldots, 2^{2022}p
   \]

3. **Check if \( n \) is southern:**
   To be southern, the differences between consecutive divisors must divide \( n \). Consider the difference \( p - 2^{2022} \):
   \[
   p - 2^{2022}
   \]
   For \( n \) to be southern, \( p - 2^{2022} \) must divide \( 2^{2022}p \). However, it also must divide:
   \[
   2^{2022}p - 2^{4044} = 2^{2022}(p - 2^{2022})
   \]
   Since \( p - 2^{2022} > 2^{4044} \), it cannot divide \( 2^{4044} \), leading to a contradiction. Therefore, \( n \) is not southern.

4. **Verify the number of southern divisors:**
   The divisors \( 2^k \) for \( k = 0, 1, 2, \ldots, 2021 \) are all southern because the differences between consecutive powers of 2 are also powers of 2, which divide \( 2^{2022}p \).

Thus, \( n = 2^{2022}p \) is a positive integer that is not southern and has exactly 2022 positive divisors that are southern.

### Part (b)
1. **Generalize the form of \( n \):**
   Any number of the form \( 2^{2022}p \), where \( p \) is a prime greater than \( 2^{2022} + 2^{4044} \), will have the same properties as described in part (a).

2. **Infinitely many such numbers:**
   There are infinitely many primes greater than \( 2^{2022} + 2^{4044} \) (by the infinitude of primes). Therefore, there are infinitely many numbers of the form \( 2^{2022}p \) that are not southern and have exactly 2022 positive divisors that are southern.

\(\blacksquare\)

The final answer is \( \boxed{ 2^{2022}p } \) where \( p \) is a prime greater than \( 2^{2022} + 2^{4044} \).