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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!)

  Submitted by Ady TZIDON    
Rating: 3.5000 (2 votes)
Solution: (Hide)
To start with, observe that if the claim holds for all integers, it holds for rational numbers as well. This permits us to focus on positive integers. For those, 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 applies to all integer numbers.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Hints/TipsSome hints....Danish Ahmed Khan2012-10-27 07:21:12
proofDaniel2010-09-22 11:08:19
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