   Chapter 8.4, Problem 27ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# In 26 and 27, use the extended Euclidean algorithm to find the greatest common divisor of the given numbers and express it as a linear combination of the two numbers. 4158 and 1568.

To determine

To calculate:

The greatest common divisor of 4158and1568 and express it in a linear combination of two numbers.

Explanation

Given information:

The given numbers are 4158and1568.

Formula used:

Division algorithm:

Let a be an integer and d be a positive integer. Then, there are unique integers qandr with 0r<d such that a=dq+r, here, q is the quotient and r is the remainder.

The greatest common divisor of two numbers aandb is the integer d which satisfies the following properties:

d|aandd|b

For all integers c, when c|aandc|b, then cd.

Calculation:

To find the greatest common divisor, use the Euclidean algorithm.

Step 1: 4158=1568×2+10221022=41581568×2

Step 2: 1568=1022×1+546546=15681022×1

Step 3: 1022=546×1+476476=1022546×1

Step 4: 546=476×1+7070=546476×1

Step 5: 476=70×6+5656=47670×6

Step 6: 70=56×1+1414=7056×1

Step 7: 56=14×4+0

So, 14 is the remainder.

gcd(4158,1568)=14

To write the gcd in a linear combination, use the step 6

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