Let $M$ be the least common multiple of $1,$ $2,$ $\dots,$ $12$, $13$, $14$, $15$, $16$. How many positive divisors does $M$ have?
LCM = 2^4 * 3^2 * 5 * 7 * 11* 13
2 = 2 9 = 3^2 16 = 2^4
3 = 3 10 = 2 * 5
4 = 2^2 11 = 11
5 = 5 12 = 2^2 * 3
6 = 2 * 3 13 = 13
7 = 7 14 = 7 * 2
8 = 2^3 15 = 3 * 5
LCM = 2^4 * 3^2 * 5 * 7 * 11 * 13
# positive divisors = (4 +1) (2 + 1) (1 + 1)^4 = 240
+1055 
