Suppose $a$ and $b$ are positive integers such that $\gcd(a,b)$ is divisible by exactly $7$ distinct primes and $\mathop{\text{lcm}}[a,b]$ is divisible by exactly $28$ distinct primes.

If $a$ has fewer distinct prime factors than $b$, then $a$ has at most how many distinct prime factors?
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 $\mathop{\text{lcm}}[a,b]$ 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.
17
Level 5
Number Theory
Problem: Suppose $a$ and $b$ are positive integers such that $\gcd(a,b)$ is divisible by exactly $7$ distinct primes and $\mathop{\text{lcm}}[a,b]$ is divisible by exactly $28$ distinct primes.

If $a$ has fewer distinct prime factors than $b$, then $a$ has at most how many distinct prime factors?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve the problem, we start by recalling the relationship between the greatest common divisor (gcd) and the least common multiple (lcm) of two integers. Specifically, we have the formula:

\[
\gcd(a, b) \times \mathop{\text{lcm}}[a, b] = a \times b
\]

Let \( p_1, p_2, \ldots, p_7 \) be the distinct primes that divide \( \gcd(a, b) \). Since \( \gcd(a, b) \) is divisible by exactly 7 distinct primes, we can express \( \gcd(a, b) \) as:

\[
\gcd(a, b) = p_1^{e_1} p_2^{e_2} \cdots p_7^{e_7}
\]

for some non-negative integers \( e_i \).

Next, let \( a \) have \( m \) distinct prime factors and \( b \) have \( n \) distinct prime factors. The distinct prime factors of \( a \) can be expressed as \( p_1, p_2, \ldots, p_7, p_8, \ldots, p_m \) where \( p_8, \ldots, p_m \) are the additional primes that divide \( a \) but not \( b \). Similarly, \( b \) can be expressed as \( p_1, p_2, \ldots, p_7, p_{8'}, \ldots, p_n \) where \( p_{8'}, \ldots, p_n \) are the additional primes that divide \( b \) but not \( a \).

The lcm of \( a \) and \( b \) is given by the union of their prime factors. Therefore, the number of distinct primes dividing \( \mathop{\text{lcm}}[a, b] \) is:

\[
\text{Number of distinct primes in } \mathop{\text{lcm}}[a, b] = m + n - 7
\]

This is because the primes in \( \gcd(a, b) \) are counted in both \( a \) and \( b \).

Given that \( \mathop{\text{lcm}}[a, b] \) is divisible by exactly 28 distinct primes, we have:

\[
m + n - 7 = 28
\]

Rearranging gives:

\[
m + n = 35
\]

Since it is stated that \( a \) has fewer distinct prime factors than \( b \), we can express this as:

\[
m < n
\]

From \( m + n = 35 \), we can substitute \( n \) with \( 35 - m \):

\[
m < 35 - m
\]

Solving this inequality:

\[
2m < 35 \implies m < 17.5
\]

Since \( m \) must be a positive integer, the maximum value for \( m \) is 17. Thus, \( a \) can have at most 17 distinct prime factors.<|END_OF_LATENT|> The final answer is $17$.