   Chapter 2.4, Problem 15E

Chapter
Section
Textbook Problem

Let r 0 = b > 0 . With the notation used in the description of the Euclidean Algorithm, use the result in Exercise 14 to prove that ( a ,   b ) = r n , the last nonzero remainder.If b > 0 and a = b q + r , prove that ( a ,   b ) = ( b ,   r ) .

To determine

To prove: Let r0=b>0 with notation used in Euclidean algorithm then (a,b)=rn, the last non-zero remainder.

Explanation

Given information:

If b>0 and a=bq+r, then (a,b)=(b,r).

Formula used:

Euclidean Algorithm:

Let a and b are integers with b>0; then, the procedure

a=bq0+r10r1<bb=r1q1+r20r2<r1r1=r2q2+r30r3<r2rk=rk+1qk+1+rk+20rk+2<rk+1

is known as Euclidean Algorithm.

Since the integers r1,r2,,rk+2 are decreasing and are all non-negative, there is a smallest integer n such that rn+1=0:

rn1=rnqn+rn+10=rn+1

If r0=b, then the last non-zero remainder rn is always the greatest common divisor of a and b.

Proof:

Let a and b are integers with b>0, therefore, by using Euclidean Algorithm,

As

a=bq0+r10r1<bb=r1q1+r20r2<r1r1=r2q2+r30r3<r2rk=rk+1qk+1+rk+2

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