Re: Greatest Common Factor
- From: Tim Smith <reply_in_group@xxxxxxxxxxxxxxxx>
- Date: Fri, 14 Nov 2008 15:32:24 -0800
In article <20081114042342.J88981@xxxxxxxxxxxxxxx>,
William Elliot <marsh@xxxxxxxxxxxxxxxx> wrote:
Apparently if a,b,n in N, then
(a,b)^n = (a^n, b^n).
Is that correct? If so, how does one prove it?
A remarkable (at least it seems so to me) result follows from that:
Let d = smallest positive integer of the form a x + b y.
Let D = smallest positive integer of the form a^n u + b^n v.
Then d^n = D.
I find that remarkable because it isn't even apparent (at least to me)
that (ax+by)^n is a linear combination of a^n, b^n. It looks like
(ax+by)^n should have all kinds of other powers of a and b. But for the
special values of x and y that minimize d, those must all add up in such
a way to get multiples of a^n or b^n.
--
--Tim Smith
.
- References:
- Greatest Common Factor
- From: William Elliot
- Greatest Common Factor
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