help!

[Jason Lam]

Hello!!

Does anyone know how to dertermine the smallest positive integer x for
which the sum x + 2x + 3x + 4x+.....+100x is a perfect square?

THANKS!

Jason Lam

[Tony2back]

In article <3151b9b4...@shts.seed.net.tw>, la...@icacomp.com (Jason
Lam) writes:

>
>Does anyone know how to dertermine the smallest positive integer x for
>which the sum x + 2x + 3x + 4x+.....+100x is a perfect square?
>
>

Write this as x(1 + 2 + 3 + .......+ 100) = x*100*101/2
= 50*101*x
= 5^2 * 2 * 101 *
x

To make this a perfect square x must equal 2 * 101 = 202

Anthony Hugh Back

[Lee Jaap]

In article <3151b9b4...@shts.seed.net.tw> la...@icacomp.com (Jason Lam) writes:

|>Does anyone know how to dertermine the smallest positive integer x for
|>which the sum x + 2x + 3x + 4x+.....+100x is a perfect square?

First, do you know how to sum an arithmetic series? One way to
represent the sum is (first + last) * (number of terms) / 2.
The part (first + last) / 2 gives an average value for all the
terms, which you then multiply by the number of terms.

Post again if this is not sufficient.
--
J Lee Jaap <Jaa...@ASMSun.LaRC.NASA.Gov> +1 804/865-7093
employed by, not necessarily speaking for,
AS&M Inc, Hampton VA 23666-1340

[Pete Vermeire]

how about 202?

On Thu, 21 Mar 1996, Jason Lam wrote:

> Hello!!

>
> Does anyone know how to dertermine the smallest positive integer x for
> which the sum x + 2x + 3x + 4x+.....+100x is a perfect square?
>

> THANKS!
>
> Jason Lam

[Michelis]

In article <3151b9b4...@shts.seed.net.tw>, la...@icacomp.com says...

>
>Hello!!
>
>Does anyone know how to dertermine the smallest positive integer x for
>which the sum x + 2x + 3x + 4x+.....+100x is a perfect square?
>
>THANKS!
>
>Jason Lam

>you mean 1x+ 2x+ .....+99x + 100x = 1050x
1050x = a^2, where a is an integer.
from this you know a must be less then or equal to 1050.
beyond that your on your own, bro!

[Martin Harvey]

In article: <4j90cu$i...@no-names.nerdc.ufl.edu> fri...@grove.ufl.edu
(Michelis) writes:
> Path:
harvey27.demon.co.uk!news.demon.co.uk!dispatch.news.demon.net!demon!btnet!ne
wsfeed.internetmci.com!usenet.eel.ufl.edu!clas.ufl.edu!usenet.ufl.edu!usenet
> From: fri...@grove.ufl.edu (Michelis)
> Newsgroups: alt.algebra.help
> Subject: Re: help!
> Date: 26 Mar 1996 14:53:18 GMT
> Organization: University of Florida
> Lines: 16
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> X-Newsreader: WinVN 0.99.5

Well... my first impulse to solve this (cheating programmer that I am) was
to write a program to solve it...

This gave the solutions as follows:

1050x=a^2 (for 0<=a<=1050)

solutions were
0
210
420
630
840
1050

This then led me into a bit of thought... the factors of 1050 are 2,5,5,and
21, and it seems fairly clear that:

2*5*5*21*x=a^2

as a result, all we have to do is to set up x so that the number on the left
is a perfect square... ie multiply by the right factors so that every factor
occurs twice. ie

2*5*5*21(21*2)=a^2

so

2*2*5*5*21*21=a^2 and thus a= 2*5*21 = 210.

so for any y*x=a^2 where y is fixed natural number, and x,a are free, it is
obvious that a can be found by.

1) Finding all the factors of y.
2) multiplying by other factors so that all the factors are repeated an even
number of times.
3) Taking the square root.

It's always easy when you know where you're going!

-------------------------------------------------------------
| Martin Harvey |
| Term Time: mc...@cam.ac.uk |
| Holidays: mc...@harvey27.demon.co.uk |
| |
| Home page: http://www.pem.cam.ac.uk:81/mch24/home.html |
-------------------------------------------------------------

[C.L. Radia]

In article <Pine.SOL.3.91.960323145951.10035D-100000@aslan>, Pete Vermeire <verm...@math.unc.edu> writes:
|>
|> how about 202?
|>
|> On Thu, 21 Mar 1996, Jason Lam wrote:
|>

|> > Hello!!
|> >
|> > Does anyone know how to dertermine the smallest positive integer x for
|> > which the sum x + 2x + 3x + 4x+.....+100x is a perfect square?
|> >
|> > THANKS!
|> >
|> > Jason Lam

Yes - the sum is 5050x = 25*202x. Now, there are no factors of 202 that are
squares. So, we must have x = 202.

It's all to do with prime factorisation of 5050.

Chetan.

[Jean-Luc Becker]

Jason Lam wrote:
>
> Hello!!
>
> Does anyone know how to dertermine the smallest positive integer x for
> which the sum x + 2x + 3x + 4x+.....+100x is a perfect square?
>
> THANKS!
>
> Jason Lam

x + 2x + 3x + 4x+.....+100x = x(1+2+3+4+ ... +100)=x * (100*101/2)=5050x.
Futhermore, 5050=2*5*5*101, where all the factors are prime, so the smallest
value for x is 2*101=202.