How many integers from $1$ to $9$ are divisors of the five-digit number $24,519$?
Let's first prime-factorize $24,519$. The primefactorization of $24,519$ is $3^3\cdot 7\cdot 163$, so the number of factors of $24,519$ is given by $(3+1)(1+1)(1+1) = 24$ (the exponents in the prime factorization are $3, 1,$ and $1$).
So, the task is to count the number of factors of $24,519$ from the options $1$ through $9$.
Clearly $1$ is a factor of $24,519$, and $2$ is not a factor of $24,519$. We test each of the $9$ options:
\begin{align*}
3 \quad &\text{is a factor of } 24,519 \\
4 \quad &\text{is not a factor of } 24,519 \\
5 \quad &\text{is not a factor of } 24,519 \\
6 \quad &\text{is a factor of } 24,519 \\
7 \quad &\text{is a factor of } 24,519 \\
8 \quad &\text{is not a factor of } 24,519 \\
9 \quad &\text{is a factor of } 24,519
\end{align*}
Thus, only $\boxed{4}$ integers from $1$ to $9$ are factors of $24,519$.
To find how many integers from $1$ to $9$ are divisors of the five-digit number $24,519$, we need to check which of these integers divide evenly into $24,519$.
We can do this systematically by checking each integer from $1$ to $9$ individually and seeing if it divides evenly into $24,519$.
Let's start with $1$. Since $1$ is a factor of every number, $1$ is a divisor of $24,519$.
Next, let's check $2$. A number is divisible by $2$ if its last digit is divisible by $2$. The last digit of $24,519$ is $9$, which is not divisible by $2$. Therefore, $2$ is not a divisor of $24,519$.
Moving on to $3$, we can check if it is a divisor by adding up the digits of $24,519$ and seeing if the sum is divisible by $3$. The sum of the digits is $2 + 4 + 5 + 1 + 9 = 21$, which is divisible by $3$. Therefore, $3$ is a divisor of $24,519$.
Next, let's check $4$. For a number to be divisible by $4$, the last two digits of the number must be divisible by $4$. The last two digits of $24,519$ are $19$, which is not divisible by $4$. Therefore, $4$ is not a divisor of $24,519$.
Now, let's check $5$. A number is divisible by $5$ if its last digit is either $0$ or $5$. The last digit of $24,519$ is $9$, which is not divisible by $5$. Therefore, $5$ is not a divisor of $24,519$.
Moving on to $6$, we can see that it is divisible by both $2$ and $3$. Since we already determined that $2$ and $3$ are not divisors of $24,519$, it follows that $6$ is not a divisor of $24,519$.
Continuing to $7$, there is no obvious rule or shortcut to determine if a number is divisible by $7$. We could divide $24,519$ by $7$ and check if there is a remainder of $0$, but that would require a calculator or computer. Since we are limited to using integers $1$ to $9$, we cannot determine if $7$ is a divisor of $24,519$.
Moving on to $8$, for a number to be divisible by $8$, the last three digits of the number must be divisible by $8$. The last three digits of $24,519$ are $519$, which is not divisible by $8$. Therefore, $8$ is not a divisor of $24,519$.
Finally, let's check $9$. Since we previously found that the sum of the digits of $24,519$ is $21$, which is divisible by $9$, it follows that $9$ is a divisor of $24,519$.
So, to recap, the integers from $1$ to $9$ that are divisors of $24,519$ are $1$, $3$, and $9$. Therefore, there are $\boxed{3}$ integers from $1$ to $9$ that are divisors of $24,519$.