Factorial Fractions

Number Theory Level 4

Calvin takes the simplest form of any rational number and multiplies its numerator and denominator. Denote $L$ as the number of rational numbers $n \in (0, 1)$ such that Calvin outputs a product of $50!$ . Find $\log_{2}{L}$ .

13 14 12 15

1 solution

Alan Yan
Sep 12, 2015

Notice how the numerator and denominator have to be relatively prime. This means that one type of prime must be either included on the top or the bottom, but not both.

$50!$ has $15$ distinct prime factors: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47$ . Each of these factors can go to two places, yielding $2^{15}$ possibilities.

However, we want the number to be less than $1$ , and the possibilities we have can be paired with their reciprocal where only one is greater than one and the other is less than one. Therefore, we need to divide by two, which yields a number of $2^{14}$ . Taking $log_{2}$ , we find that the answer is $\boxed{14}$ .

Factorial Fractions

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