Let $M$ be the least common multiple of $1,$ $2,$ $\dots,$ $12$, $13$, $14$, $15$, $16$. How many positive divisors does $M$ have?
1=1 5 = 5 9 = 3^2 13 =13
2=2 6 = 2 * 3 10 = 2*5 14= 2*7
3=3 7 = 7 11=11 15 = 3*5
4=2^2 8 = 2^3 12 = 2^2 * 3 16 = 2^4
LCM = 2^4 * 3^2 * 5 * 7 * 11 * 13 = 720720 = M
No. of Divisors = 5*3*2*2*2*2 = 240
+2653 
