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Topic: Division Algorithm Proof
Replies: 5   Last Post: Aug 16, 1999 4:29 PM

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Jon Haugsand

Posts: 26
Registered: 12/12/04
Re: Division Algorithm Proof
Posted: Aug 16, 1999 3:32 AM
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* Randy Ades
> Does anyone know a proof of the following algorithm(Division Algorithm):
> Let a and b be integers with b > 0. Then there exist unique integers q
> and r with the property that
> a = b*q + r, where r >= 0 and r < b.

Hint: Look at the set

S = {a-b*t | a-b*t non-negative and t integer }

Prove S is non-empty. From there take the smallest element of S...

Uniqueness: Assume b*q' + r' = b*q + r, then
r' - r = b * (q - q') and |r' - r| < b, and so on.

Jon Haugsand
Norwegian Computing Center,
<mailto://> Pho: +47 22852608 / +47 22852500,
Fax: +47 22697660, Pb 114 Blindern, N-0314 OSLO, Norway

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