1. What is the largest factor of 130000 that does not contain the digit 0 or 5 ? Proposed by: Farrell Eldrian Wu
Answer: 26
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 .