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Rationalize by factorials (Posted on 2010-09-22) Difficulty: 2 of 5
Prove that every positive rational number can be expressed as a quotient of products of factorials of primes.
For example, 10/9=2!*5!/(3!*3!*3!)

See The Solution Submitted by Ady TZIDON    
Rating: 3.5000 (2 votes)

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proof | Comment 1 of 2

first I will show that every prime number can be represented by a ratio of products of prime factorials
now 2=2! and 3=3!/2! and any larger prime p= p!/(p-1)! and (p-1)! can be broken down into product of smaller primes, each of which can be represented as the ratio of the product of prime factorials.

now if every prime number can be represented by the ratio of products of prime factorials then for any postive rational number x/y simply take the prime factorizations of x and y, represent each of their prime factors and then combine and simplify and you will end up with a ratio of product of prime factorials.

for example 24/25
24=2^3*3
2=2!
3=3!/2!
24=(2!)^3*(3!/2!)=2!*2!*3!=2!^2*3!
25=5^2
5=5!/4!=5!/(2!*2!*(3!/2!)*2!)=5!/(2!^2*3!)
thus
24/25 = (2!^6*3!^3)/(5!^2)

EDIT: made corrections to mistakes in my example

Edited on September 22, 2010, 4:49 pm
  Posted by Daniel on 2010-09-22 11:08:19

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