Proposition 533 Let a b c E Z 1 If a 1 then a 1 2 If a b

Proposition 5.3.3…...

Let a, b, c E Z.

1. If a | 1 then a = +- 1

2. If a | b and b | a, then a = +- b.

3. If a | b and a | c, then a | bx + cy for any x, y e Z

4. if a | b and b|c, then a|c.

-(a) Prove part 2 of proposition 5.3.3

-(b) Prove part 3 of proposition 5.3.3

-(c) Prove part 4 of proposition 5.3.3

-(d) State the contrapositive of each part of Proposition 5.3.3

-(e) State the converse of each part of Proposition 5.3.3 Prove or disprove each converse

(x) a |1 iff there exists b such that ab=1.

iff |a|=|b|=1 , iff a = 1 or -1

(a) If a |b and b|a then there exist c and d in Z such that

b =ac and a =bd which together imply

ab =abcd , which implies cd =1.

From (x) c , d are 1 or -1, implying a =b or -b.

(b) If a|b and a|c then there exist p and q such that

b=ap and c =aq

bx+cy= apx+aqy=a(px+qy) .

So a |bx+cy for any x,y in Z.

b=pa and c =qb

Then c=qb=qba which implies a |c , as required.

(d) (i) If a is neither 1 nor -1,then a does not divide 1

(ii) If a is neither b nor -b , then either a does not divide b or b does not divide a

(iii)If a does not divide bx+cy, then either a does not divide b or a does not divide c

(iv) if a does not divide c , then either a does not divide b or b does not divide c.

(e) (i) if a=1 or -1 then a divides 1 (TRUE, )

(ii) If a= b or -b , then a divides b and b divides a (TRUE)

(iii) If a divides bx+cy , then a divides b and a divides c.

FALSE: take a=2, b =1, c =-1 x,y=1. Then a |bx+cy , but 2 divides neither b nor c

(iv) If a|c then a|b and b|c.

FALSE: take a =2,c=4 and b=3