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The Nuts and Bolts of Proofs, 4th Edition
book

The Nuts and Bolts of Proofs, 4th Edition

by Antonella Cupillari
January 2011
Beginner content levelBeginner
296 pages
11h 43m
English
Academic Press
Content preview from The Nuts and Bolts of Proofs, 4th Edition
Assume that the statement is true for a generic number n > b. Then consider a = n +1.
a = n + 1 = ðbq
1
+ r
1
Þ+ 1
= bq
1
+ ðr
1
+ 1Þ
Since 0 r
1
< b,then1 r
1
+1 b.Ifr
1
+1 b,thenjustsetq
1
= q and r
1
+1=r.Ifr
1
+1=b,
we can write
a = bq
1
+ b = bq + r
where q = q
1
+ 1 and r =0.
We are now going to prove the uniqueness part of the statement. Assume that there are two pairs
of integers, q and r, q and r, such that
a = bq + r = bq + r
with q 0and0 r < b ,andq 0and0 r < b. Without loss of generality we can assume that r r
(why?).
Thus,
bðq qÞ= r r 0:
Since r > r r (why?), we have b > r r (why?). Therefore,
b > bðq qÞ 0:
Dividing by b we obtain ...
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Publisher Resources

ISBN: 9780123822178