b. Suppose that

a=bq_(1)+r_(1)=bq_(2)+r_(2)

for integers

q_(1),q_(2),r_(1)

,

r_(2)

. Prove that if either of the relations

r_(1)!=r_(2)

or

q_(1)!=q_(2)

bolds, then both must hold.\ c. Suppose that there are integers

q_(1),q_(2),r_(1),r_(2)

with

a=bq_(1)+r_(1)=bq_(2)+r_(2)r_(1)!=r_(2)bbr_(2)-r_(1)br_(2) is a positive integer less than b.\ ii. r_(2)-r_(1) is an integer multiple of b.\ Explain why the conclusions (i) and (ii) are contradictory. and conclude that there can be at most representation (*).r_(1). Show that i. r_(2) is a positive integer less than b.\ ii. r_(2)-r_(1) is an integer multiple of b.\ Explain why the conclusions (i) and (ii) are contradictory. and conclude that there can be at most representation (*).0

Suppose that a=bq1+r1=bq2+r2 for integers q1,q2,r1, r2. Prove that if either of the relations r1=r2 or q1=q2 bolds, then both must boid. c. Suppose that there are integers q1,q2,r1,r2 with 0r1