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October 4th, 2010, 10:53 AM   #1
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If n is a positive odd number, prove that 24 divides n - n

Well,

Case Base: n=3, then 27-3 = 24|24, OK!

Inductive Hypothesis: k - k divide 24. We assume True.

Then (k+1) - (k+1) ->

(k - k) + 3k + 3k.


Now how to show that 3k + 3k, or 3(k+k) also divides 24?
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October 4th, 2010, 10:58 AM   #2
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Re: If n is a positive odd number, prove that 24 divides n

I think a direct proof is easier. Note that the polynomial is n(n + 1)(n - 1).
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October 4th, 2010, 10:59 AM   #3
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Re: If n is a positive odd number, prove that 24 divides n

I wouldn't use induction.

(edit... oh, I'm late!)
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October 4th, 2010, 11:12 AM   #4
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Re: If n is a positive odd number, prove that 24 divides n

Hi guys,

Unfortunately, I must use induction.


Thanks.
If you dont mind, take a look at this topic and try to help me with this one:
viewtopic.php?f=40&t=16157
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October 4th, 2010, 11:19 AM   #5
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Re: If n is a positive odd number, prove that 24 divides n

I don't know how cheeky you're allowed to be, but since the inductive conclusion is (always) true - for reasons that we can show - then the proof "by induction" is complete.
At least that's what I would do...
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October 4th, 2010, 11:58 AM   #6
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Re: If n is a positive odd number, prove that 24 divides n

If you want to use induction, let f(n) = n^3 - n. Then calculate f(n+1) - f(n) -- you already know that f(n) is divisible by 24 by induction, so it suffices to show that f(n+1) - f(n) is.
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October 7th, 2010, 02:39 AM   #7
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Re: If n is a positive odd number, prove that 24 divides n

If you want to use induction, let f(n) = n^3 - n. Then calculate f(n+2) - f(n) -- you already know that f(n) is divisible by 24 by induction, so it suffices to show that f(n+2) - f(n) is, since n is odd.

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October 7th, 2010, 05:13 AM   #8
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Re: If n is a positive odd number, prove that 24 divides n

Yeah, what he said.
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October 7th, 2010, 12:58 PM   #9
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Quote:
Originally Posted by usermind
Then (k+1) - (k+1) . . .
In other words, k is odd, so you should have used k+2, and then the rest is easy (if you also use the fact that k+1 is even).
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October 7th, 2010, 02:12 PM   #10
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Re: If n is a positive odd number, prove that 24 divides n

Quote:
Originally Posted by CRGreathouse
Yeah, what he said.
I feel like I'm taking crazy pills!
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