SOLUTION: Let $M$ be the least common multiple of $1,$ $2,$ $\dots,$ $12$, $13$, $14$, $15$, $16$. How many positive divisors does $M$ have?

Question 1207624: Let $M$ be the least common multiple of $1,$ $2,$ $\dots,$ $12$, $13$, $14$, $15$, $16$. How many positive divisors does $M$ have?
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

M is the LCM of 1, 2, 3, ..., 15, 16

The largest number of factors of 2 in any of those numbers is 4 (2^4=16)
The largest number of factors of 3 in any of those numbers is 2 (3^2=9)
No other prime factor less than 16 occurs more than once. So

M = (2^4)(3^2)(5^1)(7^1)(11^1)(13^1)

The number of positive divisors of M is (5*3*2*2*2*2) = 240

ANSWER: 240

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