How many integers from $1$ to $9$ are divisors of the five-digit number $24,519$?

Answer 1

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 $24,519$.