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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 273 = 2424, 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: 937 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: 937 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: 937 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  
Global Moderator Joined: Dec 2006 Posts: 18,844 Thanks: 1566  Quote:
 
October 7th, 2010, 02:12 PM  #10  
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 Quote:
 

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