Factorial Products

Here is problem B1 from 70th Annual William Lowell Putnam Mathematical Competition (2010).

Show that every positive rational number can be written as a quotient of products of factorials of (not necessarily distinct) primes. For example,
 
10

9
 = 
2!·5!

3!·3!·3!

Solution

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Copyright © 1996-2017 Alexander Bogomolny

Show that every positive rational number can be written as a quotient of products of factorials of (not necessarily distinct) primes. For example,
 
10

9
 = 
2!·5!

3!·3!·3!

Solution

To start with, observe that if the claim holds for all integers, it holds for rational numbers as well. The converse is also true. This permits us to focus on positive integers. For the latter, the mathematical induction seems a natural way to proceed. We apply induction to show that every prime can be represented as claimed. This is true for 2 = 2!. Suppose the claim holds for all the primes less than the given prime p > 2. Since
 
p
 = 
p!

(p - 1)!

and (p - 1)! admits a factorization into a product of primes smaller than p, we see, by the induction hypothesis, that the claim holds for p as well and so holds for all prime numbers.

Now, since every integer is subject to a prime factorization, and every prime has been shown to be in the required form, the same holds for every integer.


Related material
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  • Permutations
  • Long Runs with No Primes
  • Partitioning a Circle
  • Binomial Coefficient
  • Patterns in Pascal's Triangle
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