Help finding the error in the proof

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tonyc1970
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Joined: Fri Aug 02, 2013 4:58 pm
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Help finding the error in the proof

Hi,

First let me say that I am not looking for the answer, just a few hints to help me get started. I want to make sure I understand this as much as possible.

The problem states for me to identify the error in the proof:

Let u, m, n be three integers. If u|mn and gcd(u,m) = 1, then m = +- 1. If gcd(u,m ) = 1, then 1 = us + mt for some integers s,t. If u|mn, then us = um for some integer s. Hence, 1 = mn + mt = m(s + t), which implies that m|1, and therefor m = +- 1.

Thanks for any help,

Tony

nona.m.nona
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Re: Help finding the error in the proof

...identify the error in the proof:

Let u, m, n be three integers. If u|mn and gcd(u,m) = 1, then m = +- 1. If gcd(u,m ) = 1, then 1 = us + mt for some integers s,t. If u|mn, then us = um for some integer s. Hence, 1 = mn + mt = m(s + t), which implies that m|1, and therefor m = +- 1.
The proof makes little sense, as it appears to start off by stating what it then concludes. What statement is this proof meant to be proving?

Also, what have you used in your attempts to find errors? What progress have you made? Kindly please be complete. Thank you.

tonyc1970
Posts: 13
Joined: Fri Aug 02, 2013 4:58 pm
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Re: Help finding the error in the proof

Hi nona.m.nona,

This is the proof exactly as it is written in my book,

Code: Select all

`Let u, m, n be three integers. If u|mn and gcd(u,m) = 1, then m = (+ or -) 1. If gcd(u,m ) = 1, then 1 = us + mt for some integers s,t. If u|mn, then us = um for some integer s. Hence, 1 = mn + mt = m(s + t), which implies that m|1, and therefore m = (+ or -) 1.`
I am to find the error in this proof but I am not sure where to start.

Thanks

nona.m.nona
Posts: 288
Joined: Sun Dec 14, 2008 11:07 pm
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Re: Help finding the error in the proof

This is the proof exactly as it is written in my book,

Code: Select all

`Let u, m, n be three integers. If u|mn and gcd(u,m) = 1, then m = (+ or -) 1. If gcd(u,m ) = 1, then 1 = us + mt for some integers s,t. If u|mn, then us = um for some integer s. Hence, 1 = mn + mt = m(s + t), which implies that m|1, and therefore m = (+ or -) 1.`
You present the above as the proof. But of what is this the proof? What is the exact statement of the proposed theorem for which the above is the proof?

tonyc1970
Posts: 13
Joined: Fri Aug 02, 2013 4:58 pm
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Re: Help finding the error in the proof

This is the exact statement, as I have already pointed out. I cannot provide any more information since this is all I have.

Code: Select all

`Let u, m, n be three integers. If u|mn and gcd(u,m) = 1, then m = (+ or -) 1. If gcd(u,m ) = 1, then 1 = us + mt for some integers s,t. If u|mn, then us = um for some integer s. Hence, 1 = mn + mt = m(s + t), which implies that m|1, and therefore m = (+ or -) 1.`

stapel_eliz
Posts: 1628
Joined: Mon Dec 08, 2008 4:22 pm
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This is the exact statement, as I have already pointed out. I cannot provide any more information since this is all I have.
Is it possible that the thing you're trying to prove is the first bit? And then the rest is the attempted proof?
"Theorem": Let u, m, n be three integers. If u|mn and gcd(u,m) = 1, then m = (+ or -) 1.

"Proof": If gcd(u,m ) = 1, then 1 = us + mt for some integers s,t. If u|mn, then us = um for some integer s. Hence, 1 = mn + mt = m(s + t), which implies that m|1, and therefore m = (+ or -) 1.
The first bit (in blue above) is clearly not true. It makes more sense, I think, if one assumes the green part (above) is the "proof" part.

My suggestion would be to attack the second statement in the green part.