, 2018

Solution:

If the number is a multiple of $5$, then its units digit will be either $0$ or $5$. Hence, the largest such number must have no factors of $5$.

We have $130000 = 2^{4} \cdot 5^{4} \cdot 13$. Removing every factor of $5$, we get that our number must be a factor of $2^{4} \cdot 13 = 208$.

If our number contains a factor of $13$, we cancel the factor of $2$ from $208$, $104$, and $52$ until we get $26$. Otherwise, the largest number we can have is $2^{4} = 16$. We conclude that the answer is $26$.