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**Stanley_Marsh****Member**- Registered: 2006-12-13
- Posts: 345

Here's my proof:

*Last edited by Stanley_Marsh (2007-04-12 14:07:11)*

Numbers are the essence of the Universe

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

You're not using what your given. Let |ab| = k. Then (ab)^k = a^k * b^k = 1. Thus, it must be that |a| | k and |b| | k, and k is the least of such numbers. Thus, k = lcm(n, m). But (m, n) = 1, so lcm(n, m) = mn.

I can guarantee your proof is invalid as you didn't use the fact that ab = ba, and this statement is not true in general.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**Stanley_Marsh****Member**- Registered: 2006-12-13
- Posts: 345

Awww, I dont really understand the importance of commutative in this case?

Numbers are the essence of the Universe

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

(ab)^k = a^k * b^k if ab = ba.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**Zhylliolom****Real Member**- Registered: 2005-09-05
- Posts: 412

To explain Ricky's statement a little more, note that

We cannot rearrange these terms into a[sup]k[/sup]b[sup]k[/sup] if a and b do not commute. Do you see this?

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

Stanley_Marsh wrote:

You are just assuming that the *a[sup]i[/sup]* and *b[sup]j[/sup]* are distinct if 0 < *i* < *m*, 0 < *j* < *n*. Shouldnt you actually show that they are?

*Last edited by JaneFairfax (2007-04-12 16:49:48)*

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**Stanley_Marsh****Member**- Registered: 2006-12-13
- Posts: 345

I see how commutative works~

Jane: Can I do that through proving that (a,b)=1 ?

Numbers are the essence of the Universe

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

What do you mean by (*a*,*b*)?

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**Stanley_Marsh****Member**- Registered: 2006-12-13
- Posts: 345

I mean , a ,b are relatively prime

Numbers are the essence of the Universe

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

*a* and *b* are elements in the group.

You mean make use of gcd(|*a*|,|*b*|)=1? Of course. In fact, you have to use that condition.

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**Stanley_Marsh****Member**- Registered: 2006-12-13
- Posts: 345

Yeah , that's what I meant.

Numbers are the essence of the Universe

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