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        riddles >> medium >> number of divisors
        
(Message started by: Christine on Jun 3^rd, 2013, 9:59am)

Title: number of divisors
Post by Christine on Jun 3^rd, 2013, 9:59am

The number of positive divisors of 9 is equal to the square root of 9

1 is the trivial solution.

How do you prove that (1, 9) are the only solutions?

Title: Re: number of divisors
Post by rmsgrey on Jun 4^th, 2013, 7:00am

on 06/03/13 at 09:59:48, Christine wrote:

The number of positive divisors of 9 is equal to the square root of 9

1 is the trivial solution.

How do you prove that (1, 9) are the only solutions?

1) [hide]Any square has an odd number of divisors, so any solution must be an odd square.[/hide]

2) [hide]Any solution not divisible by 3 gives rise to another solution when multiplied by 9 since that multiplies both the square root and the number of divisors by 3.[/hide]

3) [hide]Any solution divisible by 3 will not give rise to another solution when multiplied by 9 since that multiplies the square root by 3, and the number of divisors by at most 5/3.[/hide]

4) [hide]Multiplying any solution by p², where p>3 and p is prime, multiplies the square root by p and the number of divisors by at most 3 so does not give rise to another solution.[/hide]

5) [hide]Multiplying any square with a square root greater than its number of divisors by an odd prime squared will never give a solution since the square root is multiplied by at least 3 and the number of divisors by at most 3.[/hide]

6) [hide]Any odd square is the product of squares of odd primes, so can be reached from 1 by repeatedly multiplying by an odd prime squared - since 1 is a solution, from the above, as soon as you multiply by 9 a second time, or by any other odd prime squared, the number produced is not a solution, and no subsequent steps will produce a solution.[/hide]