jmc

Since we know the number must be a multiple of $15$, we can eliminate $A$. We also know that after $12$ days, the students can't find any more arrangements, meaning the number has $12$ factors. Now, we just list the factors of every number, starting with $30$: $$30=1\cdot 30,2\cdot 15,3\cdot 10,5\cdot 6$$ $$60=1\cdot 60,2\cdot 30,3\cdot 20,4\cdot 15,5\cdot 12,6\cdot 10$$ $60$ has $12$ factors, so the answer is $\boxed{60}$. Note - A faster way to find the factors of a number is really helpful in competitions, as you don't want to waste time listing them all. What we can do is find the prime factorization of a number. We'll use $60$ as an example here, so we take $60=2\cdot 2\cdot 3\cdot 5$ or $60=2^2\cdot 3^1\cdot 5^1$. Now, we just add one to the power of each factor. So since we have powers of $2$, $1$, and $1$, we turn those into $3$, $2$, and $2$. Finally, we just multiply them together to get $3\cdot 2\cdot 2=12$ as the amount of factors in 60.