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#1 2007-09-18 21:39:38

Identity
Member
Registered: 2007-04-18
Posts: 934

Find suitable integers

Find integers a, b, d, s, t such that all of the following hold
(1) a > 0, b > 0,
(2) d = sa + tb, and
(3) d =/= gcd(a, b).

I think that d cannot exist, since, by Bezout's lemma, (2)

~(3). I was just wondering if I was right?

Last edited by Identity (2007-09-18 23:32:37)

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#2 2007-09-18 23:13:01

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Find suitable integers

(2) restricts d to be a multiple of the gcd, but not necessarily the gcd itself.

There are infinite solutions to this.
One would be a=2, b=3, d=5, s=t=1.


Why did the vector cross the road?
It wanted to be normal.

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#3 2007-09-18 23:31:06

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

Re: Find suitable integers

As a simple example, you can take a = b = 1, and s and t any positive integers you like.

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#4 2007-09-20 00:45:11

Identity
Member
Registered: 2007-04-18
Posts: 934

Re: Find suitable integers

i forgot to say thanks, thanks!up

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