### Part (a)
1. **Define the function $f(n)$:**
$$f(n) = \text{lcm}(1, 2, \ldots, n)$$
where $\text{lcm}$ denotes the least common multiple.

2. Observation:
For any integer $n$, $f(n)$ is the least common multiple of the first $n$ positive integers. If we consider $f(n)$ and $f(n+1)$, we note that $f(n+1)$ is the least common multiple of the first $n+1$ integers.

3. Key Insight:
If $n+1$ is not a prime number or a power of a prime, then $f(n+1) = f(n)$. This is because the least common multiple of the first $n$ integers already includes all the prime factors of $n+1$.

4. **Constructing $k$ consecutive integers:**
To find $k$ consecutive integers where $f$ is constant, we need $k$ consecutive integers none of which are primes or powers of primes. For example, consider the sequence starting from $n = 2$:
$$4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, \ldots$$
These numbers are not primes or powers of primes. We can always find such sequences of any length $k$.

5. Conclusion:
For any $k$, there exist $k$ consecutive positive integers on which $f$ is constant.

### Part (b)
1. Observation:
$f(n+1) > f(n)$ if and only if $n+1$ introduces a new prime factor or a higher power of an existing prime factor that was not already included in $f(n)$.

2. Strictly Increasing Sequence:
For $f(n)$ to be strictly increasing over a set of consecutive integers, each integer in the set must introduce a new prime factor or a higher power of an existing prime factor. This happens when each integer in the set is a prime or a power of a prime.

3. Maximum Cardinality:
The maximum possible cardinality of such a set is 2. This is because if we have three consecutive integers, at least one of them will not be a prime or a power of a prime (since primes and their powers are not consecutive except for the trivial case of 2 and 3).

4. Example Sets:
The sets $\{2, 3\}$ and $\{3, 4\}$ are examples where $f$ is strictly increasing:
$$f(2) = \text{lcm}(1, 2) = 2$$
$$f(3) = \text{lcm}(1, 2, 3) = 6$$
$$f(4) = \text{lcm}(1, 2, 3, 4) = 12$$

The final answer is $\boxed{2}$ for the maximum possible cardinality of a set of consecutive positive integers on which $f$ is strictly increasing.