Introduction The Divisibility Relation

Divisibility AND greatest COMMON DIVISORSKEITH will begin with a review of Divisibility among integers, mostly to set some notationand to indicate its properties. Then we will look at two important theorems involvinggreatest common divisors: Euclid s algorithm and Bezout s set of integers is denotedZ(from the German word Zahl = number). Divisibility RelationDefinition integers, we sayadividesbifb=akfor somek then writea|b(read as adividesb ).Example have 2|6 (because 6 = 2 3), 4|( 12), and 5|0. We have 1|bforeveryb Z. However, 6 does not divide 2 and 0 does not divide is a Relation , much like inequalities. In particular, the Relation 2|6 isnotthenumber 3, even though 6 = 2 3. Such an error would be similar to the mistake of confusingthe Relation 5<9 with the number 9 Divisibility is not symmetric: ifa|b, it is usually not true thatb|a, so you shouldnot confuse the roles ofaandbin this Relation : 4|20 but the definition ofa|bas given in Definition , and not in theform bais an integer.

DIVISIBILITY AND GREATEST COMMON DIVISORS 5 Therefore Bezout’s identity is satis ed with the integers x = 5 and y = 2442 (not y = 2442). Example 3.7.

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