My Math Forum  

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

Number Theory Number Theory Math Forum


Reply
 
LinkBack Thread Tools Display Modes
April 28th, 2013, 03:02 AM   #1
Senior Member
 
Joined: Feb 2013

Posts: 114
Thanks: 0

Number system - proving 9 digit number not divisible by 5

Prove that there is no such 9 digit number in which every digit except zero occurs (once) and which ends in 5 can't be square ( digits are not to be repeated).




Please suggest the solution of this question.
sachinrajsharma is offline  
 
April 28th, 2013, 04:27 AM   #2
Math Team
 
Joined: Apr 2012

Posts: 1,579
Thanks: 22

Re: Number system - proving 9 digit number not divisible by

Quote:
Originally Posted by sachinrajsharma
Prove that there is no such 9 digit number in which every digit except zero occurs (once) and which ends in 5 can't be square ( digits are not to be repeated).




Please suggest the solution of this question.
I don't see the connection between your subject line and the question you go on to ask.
johnr is offline  
April 28th, 2013, 04:34 AM   #3
Math Team
 
mathbalarka's Avatar
 
Joined: Mar 2012
From: India, West Bengal

Posts: 3,871
Thanks: 86

Math Focus: Number Theory
Re: Number system - proving 9 digit number not divisible by

Maybe he asks to prove that no 9 digit number follows these properties :

1) It is divisible by 5
2) Every integer from 1-9 are it's digits
3) The number is a perfect square

I haven't ran a brute-force check yet but my heuristics says that if such number exist, then it must have the last two digits 2 & 5.
mathbalarka is offline  
April 28th, 2013, 04:59 AM   #4
Math Team
 
mathbalarka's Avatar
 
Joined: Mar 2012
From: India, West Bengal

Posts: 3,871
Thanks: 86

Math Focus: Number Theory
Re: Number system - proving 9 digit number not divisible by

This is the possible list of such numbers I found using brute-force search :

100500625,102515625,104550625,106605625,108680625, 110775625,112890625,115025625,
117180625,119355625,121550625,123765625,126000625, 128255625,130530625,132825625,
135140625,137475625,139830625,142205625,144600625, 147015625,149450625,151905625,
154380625,156875625,159390625,161925625,164480625, 167055625,169650625,172265625,
174900625,177555625,180230625,182925625,185640625, 188375625,191130625,193905625,
196700625,199515625,202350625,205205625,208080625, 210975625,213890625,216825625,
219780625,222755625,225750625,228765625,231800625, 234855625,237930625,241025625,
244140625,247275625,250430625,253605625,256800625, 260015625,263250625,266505625,
269780625,273075625,276390625,279725625,283080625, 286455625,289850625,293265625,
296700625,300155625,303630625,307125625,310640625, 314175625,317730625,321305625,
324900625,328515625,332150625,335805625,339480625, 343175625,346890625,350625625,
354380625,358155625,361950625,365765625,369600625, 373455625,377330625,381225625,
385140625,389075625,393030625,397005625,401000625, 405015625,409050625,413105625,
417180625,421275625,425390625,429525625,433680625, 437855625,442050625,446265625,
450500625,454755625,459030625,463325625,467640625, 471975625,476330625,480705625,
485100625,489515625,493950625,498405625,502880625, 507375625,511890625,516425625,
520980625,525555625,530150625,534765625,539400625, 544055625,548730625,553425625,
558140625,562875625,567630625,572405625,577200625, 582015625,586850625,591705625,
596580625,601475625,606390625,611325625,616280625, 621255625,626250625,631265625,
636300625,641355625,646430625,651525625,656640625, 661775625,666930625,672105625,
677300625,682515625,687750625,693005625,698280625, 703575625,708890625,714225625,
719580625,724955625,730350625,735765625,741200625, 746655625,752130625,757625625,
763140625,768675625,774230625,779805625,785400625, 791015625,796650625,802305625,
807980625,813675625,819390625,825125625,830880625, 836655625,842450625,848265625,
854100625,859955625,865830625,871725625,877640625, 883575625,889530625,895505625,
901500625,907515625,913550625,919605625,925680625, 931775625,937890625,944025625,
950180625,956355625,962550625,968765625,975000625, 981255625,987530625,993825625,
1000140625,1006475625,1012830625,1019205625,

Everyone of these has either two 5's or a 0. Hence, such number don't exist. I think a rigorous proofs follows from the fact that the last two digits are 2 and 5.
mathbalarka is offline  
April 28th, 2013, 12:29 PM   #5
Math Team
 
Joined: Apr 2012

Posts: 1,579
Thanks: 22

Re: Number system - proving 9 digit number not divisible by

Well, it SOUNDS like he wants wants only nine digit numbers where all of the digits 1,2,3,4,5,6,7,8,9 occur. No zeroes anywhere.

Now, If all digits one through 9 occur at least once, then they all occur ONLY once and no special stipulation is required.

Obviously, all (and only) those permutations of 1,2,3,4,6,7,8 and 9 with 5 tacked on the end WILL be divisible by 5.

I can't process what "can't be square" means. If you are ruling out cases where the resultant nine digit number is a square, you will still end up with lots that meet the criteria, including the "first" case: 123467895, whose square root is greater than 111 and less than 112.

So, I am still missing the drift OR the desired result is simply wrong.

So I suppose Balarka's interpretation must be the correct one. What is desired is a proof that none of the numbers that ARE divisible by 5 are square. If you ARE a square and divisible by 5, then you must be divisible by 25. In THAT case, Balarka's brute force demo that all nine digit multiples of 25 either contain 0s or repeats, and specifically repeats of 5, which is kind of intriguing, does indeed prove the case.
johnr is offline  
April 28th, 2013, 08:43 PM   #6
Math Team
 
agentredlum's Avatar
 
Joined: Jul 2011
From: North America, 42nd parallel

Posts: 3,179
Thanks: 180

Re: Number system - proving 9 digit number not divisible by

I agree so far with everything but would like to add,

Every number that can be made using all digits 1 - 9 is divisible by 9 since the digital root of any such number is 9

Now, we are looking for a square that is also divisible by 5. A square divisible by 5 must be divisible by 25

so we are looking for a number of the form



for some integer

740 < k < 2096

all even integers can be excluded at once since the 15 will make last digit zero

Where to go next, i'm not sure.

[color=#0000BF]mathbalarka[/color], I like your approach, numbers < 123456789 or numbers > 987654321 can be crossed off your list. That eliminates 16 candidates, a slight improvement but not much.

agentredlum is offline  
April 29th, 2013, 03:37 AM   #7
Math Team
 
mathbalarka's Avatar
 
Joined: Mar 2012
From: India, West Bengal

Posts: 3,871
Thanks: 86

Math Focus: Number Theory
Re: Number system - proving 9 digit number not divisible by

Quote:
Originally Posted by agentredlum
so we are looking for a number of the form


This property is an interesting one, although it's not useful for brutal methods. I have to think a bit more to derive a rigorous proof I guess...

Quote:
Originally Posted by agentredlum
numbers < 123456789 or numbers > 987654321 can be crossed off your list. That eliminates 16 candidates
Yes, I didn't noticed this property, nice one agentredlum! This reduces the amount of checking needed to 200 candidates.
mathbalarka is offline  
April 29th, 2013, 05:49 AM   #8
Math Team
 
Joined: Apr 2012

Posts: 1,579
Thanks: 22

Re: Number system - proving 9 digit number not divisible by

Ok, building on agent's astute observation: Since whatever factors beyond 25 and 9 the number has must be odd squares, the number must ultimately be of the form:

n = 900(x^2+x)+225

So, if you plug in any n of the relevant form into the following quadratic equation:

x^2 + x + ((n-225)/900) = 0

Do you get any integral solutions? If not, then there is no such number.

Still a bit brutish, however, ...
johnr is offline  
Reply

  My Math Forum > College Math Forum > Number Theory

Tags
digit, divisible, number, proving, system



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Four-digit number life24 Number Theory 8 February 20th, 2014 11:16 AM
Four-digit number life24 Advanced Statistics 2 February 17th, 2014 05:24 AM
HELP! probability that a 7 digit number is divisible by 7? lincoln40113 Advanced Statistics 18 October 14th, 2013 11:52 PM
A=abc=a !+b !+c ! (A is a 3-digit number) Albert.Teng Algebra 10 November 8th, 2012 12:25 PM
ten-digit number Albert.Teng Algebra 6 July 21st, 2012 11:30 PM





Copyright © 2017 My Math Forum. All rights reserved.