8n divides (4n) Proof by induction

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Homework Help Overview

The discussion revolves around proving that \(8n\) divides \((4n)!\) using mathematical induction. The participants are exploring the necessary steps and reasoning involved in this proof.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the induction steps, specifically addressing the transition from \(k\) to \(k+1\). Questions are raised about the relationship between \(8k\) and \((4k)!\), and whether \(8^k\) divides \((4k)!\). There is also an exploration of simplifying \((4(k + 1))!\) in terms of \((4k)!\).

Discussion Status

The discussion is active, with participants providing insights and questioning assumptions. Some guidance has been offered regarding the proof structure, and there appears to be a recognition of the necessary details for completing the proof.

Contextual Notes

Participants are working under the constraints of a proof by induction and are examining the implications of divisibility in the context of factorials.

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Homework Statement


I need to do this by proof by induction
8n divides (4n)!

Homework Equations





The Attempt at a Solution


I already did it for 1, now I need to do for k and k+1
(4k)!t=8k
 
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If 8k divides (4k)!, how can 8k be a multiple of (4k)! as you wrote?
 
[tex]8^{k}[/tex] divides (4k)!
[tex]8^{k}[/tex]*t=(4k)!
 
Simplify (4(k + 1))! in terms of (4k)!
 
(4k+4)!=(4k+4)(4k+3)(4k+2)(4k+1)(4k)!
 
You already know that [tex]8^k[/tex] divides (4k)!, so does 8 divide everything else?
 
Yes...
 
Then you have all the details for the complete proof.
 
Ok, I see...
 

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