jmc

The prime factors of $\gcd (a,b)$ are precisely the prime factors which are common to $a$ and $b$ (i.e., the primes that divide both). The prime factors of $\mathrm{lcm}[a,b]$ are the primes which divide at least one of $a$ and $b$.

Thus, there are $7$ primes which divide both $a$ and $b$, and $28-7=21$ more primes which divide exactly one of $a$ and $b$. Since $a$ has fewer distinct prime factors than $b$, we know that fewer than half of these $21$ primes divide $a$; at most, $10$ of these primes divide $a$. So, $a$ has at most $7+10=\boxed{17}$ distinct prime factors.