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Topic: number of divisors (Read 2152 times) |
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Christine
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number of divisors
« on: Jun 3rd, 2013, 9:59am » |
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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?
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rmsgrey
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Re: number of divisors
« Reply #1 on: Jun 4th, 2013, 7:00am » |
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on Jun 3rd, 2013, 9:59am, 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?
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1) Any square has an odd number of divisors, so any solution must be an odd square.
2) 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.
3) 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.
4) 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.
5) 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.
6) 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.
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