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

 Jun 24, 2024
 #1
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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 

 

cool cool cool

 Jun 24, 2024

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