My Math Forum  

Go Back   My Math Forum > College Math Forum > Number Theory

Number Theory Number Theory Math Forum


Thanks Tree1Thanks
  • 1 Post By romsek
Reply
 
LinkBack Thread Tools Display Modes
November 9th, 2017, 10:41 AM   #1
Senior Member
 
Joined: Dec 2015
From: Earth

Posts: 177
Thanks: 23

Natural set

Show that $\displaystyle (5n)! \; \vdots \; 40^n n! \; \; ,\, n \in N$
or $\displaystyle \frac{(5n)!}{40^n n!} \in N$
If can't post proof, maybe we can (use induction).

Last edited by skipjack; November 9th, 2017 at 04:19 PM.
idontknow is offline  
 
November 9th, 2017, 11:16 AM   #2
Global Moderator
 
Joined: Dec 2006

Posts: 18,155
Thanks: 1422

What happens when $n$ = 1?
skipjack is offline  
November 9th, 2017, 11:52 AM   #3
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: Southern California, USA

Posts: 1,604
Thanks: 817

Quote:
Originally Posted by skipjack View Post
What happens when $n$ = 1?
$5! = 120$

$40^1 = 40$

$1! = 1$

$\dfrac{120}{40}=3 \in \mathbb{N}$

When attempting induction you get to a 4th degree polynomial in $n$ with a bunch of integer coefficients divided by 8. I wasn't able to proceed any further. Maybe someone else can succeed.
romsek is offline  
November 9th, 2017, 07:29 PM   #4
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: Southern California, USA

Posts: 1,604
Thanks: 817

ok.. I leave you to turn this into a formal proof but here's an example of what's going on

Let $n=3$

$(5n)! = (15)(14) \dots (4)(3)(2)(1)$

$\dfrac{(5n)!}{n!} = (15)(14) \dots (4)$

Now in those factors we have $(8,5),(4,10),(6,12,15)$ each of which multiplies to $40$. That gets you your $40^3$ multiplied by the unused factors.

If $n=4$

$(5n)! = (20)(19) \dots (5)$

we have the same tuples as before to get $40^3$ and now we have

$(20)(19)\dots (16)$ to extract another factor of $40$ out of. One such tuple is $(16,20)$ There are others.

It shouldn't be to hard to frame this as induction. You just need to show there exists a combination of factors $\in [5n+1,5(n+1)]$ that multiply to a multiple of $40$
Thanks from topsquark
romsek is offline  
November 10th, 2017, 03:04 AM   #5
Senior Member
 
Joined: Dec 2015
From: Earth

Posts: 177
Thanks: 23

This is the way I made the Set , but I don't know how to get there mathematically.

Last edited by skipjack; November 10th, 2017 at 06:10 AM.
idontknow is offline  
November 10th, 2017, 06:04 PM   #6
Member
 
Joined: Jan 2016
From: Athens, OH

Posts: 58
Thanks: 34

Legendre's formula provides an easy proof without induction:

johng40 is offline  
Reply

  My Math Forum > College Math Forum > Number Theory

Tags
natural, set



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Log / Natural Log lenoramarie Pre-Calculus 2 November 1st, 2014 05:09 AM
Natural Log mathkid Calculus 4 August 27th, 2012 10:59 AM
using natural log tsl182forever8 Calculus 2 March 1st, 2012 07:16 PM
natural numbers from sets....not very natural jinjouk Number Theory 12 June 3rd, 2008 07:11 AM
using natural log tsl182forever8 Algebra 1 December 31st, 1969 04:00 PM





Copyright © 2017 My Math Forum. All rights reserved.