Problem:
The notation (read " factorial") is defined as the product of the first positive integers. (For example, .) Define the superfactorial of a positive integer, denoted by , to be the product of the factorials of the first integers. (For example, .) How many factors of appear in the prime factorization of , the superfactorial of
Answer Choices:
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Solution:
We can write as . We can count how many times each number from to will appear in a factorial. For example, only shows up in , but shows up twice, once in and once in . Continuing this pattern, we can see that any number appears in all factorials from to , or exactly times. Thus, we can rewrite the factorial product as:
To find the total number of factors of that appear in the prime factorization of , we only need to look at the terms that are multiples of : . Each of those will contribute one factor of to the final count, except for where each contributes two factors of to the final count since . So to find how many factors of appear in the prime factorization of , we just add the exponents on the multiples of (and twice the exponent on ).
The problems on this page are the property of the MAA's American Mathematics Competitions