My Math Forum If n is a positive odd number, prove that 24 divides n³ - n
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 October 4th, 2010, 10:53 AM #1 Newbie   Joined: Mar 2010 Posts: 23 Thanks: 0 If n is a positive odd number, prove that 24 divides n³ - n Well, Case Base: n=3, then 27-3 = 24|24, OK! Inductive Hypothesis: k³ - k divide 24. We assume True. Then (k+1)³ - (k+1) -> (k³ - k) + 3k² + 3k. Now how to show that 3k² + 3k, or 3(k²+k) also divides 24?
 October 4th, 2010, 10:58 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: If n is a positive odd number, prove that 24 divides n³ I think a direct proof is easier. Note that the polynomial is n(n + 1)(n - 1).
 October 4th, 2010, 10:59 AM #3 Global Moderator     Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4 Re: If n is a positive odd number, prove that 24 divides n³ I wouldn't use induction. (edit... oh, I'm late!)
 October 4th, 2010, 11:12 AM #4 Newbie   Joined: Mar 2010 Posts: 23 Thanks: 0 Re: If n is a positive odd number, prove that 24 divides n³ 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
 October 4th, 2010, 11:19 AM #5 Global Moderator     Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4 Re: If n is a positive odd number, prove that 24 divides n³ 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...
 October 4th, 2010, 11:58 AM #6 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: If n is a positive odd number, prove that 24 divides n³ 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.
 October 7th, 2010, 02:39 AM #7 Math Team   Joined: Apr 2010 Posts: 2,778 Thanks: 361 Re: If n is a positive odd number, prove that 24 divides n³ 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. Hoempa
 October 7th, 2010, 05:13 AM #8 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: If n is a positive odd number, prove that 24 divides n³ Yeah, what he said.
October 7th, 2010, 12:58 PM   #9
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Quote:
 Originally Posted by usermind Then (k+1)³ - (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).

October 7th, 2010, 02:12 PM   #10
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Re: If n is a positive odd number, prove that 24 divides n³

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

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