Hi guys,

I need to show that for x>=1 x^3-x^5 is divisible by 12.

Not really sure how to start! :-s

Many thanks in advance. x

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- March 16th 2009, 10:17 AMAAMDivisibility Proof
Hi guys,

I need to show that for x>=1 x^3-x^5 is divisible by 12.

Not really sure how to start! :-s

Many thanks in advance. x - March 16th 2009, 12:20 PMjosh_amsterdam
The trick you often use for these type of questions is proving that (x+1) and (x) give the same remainder.

Easy example: 4^x is always divisible by 4. You take (x+1) which is: 4^(x+1) = 4*4^x. When dividing with 4 you can throw out the 4 and you're left with 4^x. Now if you prove that x=1 is divisible by four, you've also proven it for x+1 and x+2 etc.

Your problem is a little harder ofcourse, and sometimes it's easier to use (x+2) or (x+3) instead of (x+1), but you'll just have to try a little. Hope this helped, and if you still can't solve it I'll try to give you a start. ;) - March 16th 2009, 02:45 PMAAM
Thanks josh_amsterdam :-)

.... but I still don't know where to begin! :-s lol! - March 16th 2009, 05:02 PMSoroban
Hello, AAM!

That's a silly way to write the problem.

Are they deliberately being annoying?

Quote:

Show that is divisible by 12 for

Then: .

Let's ignore the leading minus-sign.

With 3 consecutive integers, one of them is a multiple of 3.

. . Hence, is divisible by 3.

There are two cases: .

. . Therefore, is divisible by 12.

- March 16th 2009, 06:11 PMAAM
Thank you Soroban! :-D