Prove that, for any positive integer

is divisible by 9.

denotes the integer part of .

I wasn't sure how to go about this question, any help?

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- Aug 26th 2010, 03:14 AMCuriosityCabinetDivisibility by 9
Prove that, for any positive integer

is divisible by 9.

denotes the integer part of .

I wasn't sure how to go about this question, any help? - Aug 26th 2010, 03:26 AMCuriosityCabinet
Please delete. Posted in wrong forum. Sorry :P

- Aug 26th 2010, 03:36 AMsa-ri-ga-ma
Check whether the given problem is divisible by 9 for n = 4 onwards.

- Aug 26th 2010, 03:38 AMCuriosityCabinet
Yes it is. But it comes to 0 for n=1,2,3.

- Aug 26th 2010, 07:30 AMchiph588@
- Aug 28th 2010, 05:18 AMmelese
Calculate .

Any integer can be written in the form , where or . It follows that .

Now, simplifying, .

Consider three cases:

1. : and since is divisible by 3, is divisible by 9.

2. : , where and . Then is divisible by 9.

3. : , and since it follows that is divisible by 9.

For cases 1. and 2. I noticed that 1 and 2 are the roots, respectively, of the polynomials and then used Polynomial Division.