jmc

Question

How many different primes appear in the prime factorization of 20! (20 factorial)? (Reminder: The number n! is the product of the integers from 1 to n. For example, 5!=5 432 1= 120.)

Accepted Answer

When we multiply the numbers from 1 to 20, we include every prime less than 20, and no numbers with prime factors greater than 20. So, the prime factorization of

20!

includes all primes less than 20, and no other primes.

The primes that fit this are

{2, 3, 5, 7, 11, 13, 17, 19}

, and there are

8

of them.