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Use the Euclidean algorithm to finda) gcd(1,

ISBN: 9780073383095 37

Solution for problem 32E Chapter 4.3

Discrete Mathematics and Its Applications | 7th Edition

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Discrete Mathematics and Its Applications | 7th Edition

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Problem 32E

Problem 32E

Use the Euclidean algorithm to find

a) gcd(1, 5).

b) gcd(100,101).

c) gcd(123,  277).

d) gcd(1529, 14039).

e) gcd( 1529, 14038).

f) gcd( 11111, 111111).

Step-by-Step Solution:
Step 1 of 3

Solution:-

Step1

Given that

We have to use the Euclidean algorithm to find gcd.

Step2

a) gcd(1, 5)

By using  Euclidean algorithm

As 1 is the last nonzero remainder.

Therefore,  gcd(1, 5) is 1.

Step3

b) gcd(100,101)

By using  Euclidean algorithm

As 1 is the last nonzero remainder.

Therefore,  gcd(100, 101) is 1.

Step4

c) gcd(123,  277)

By using  Euclidean algorithm

As 1 is the last nonzero remainder.

Therefore,   gcd(123,  277) is 1.

Step5

d) gcd(1529, 14039)

By using  Euclidean algorithm

As 139 is the last nonzero remainder.

Therefore,  gcd(1529, 14039) is 139.

Step6

e) gcd( 1529, 14038)

By using  Euclidean algorithm

As 1 is the last nonzero remainder.

Therefore, gcd( 1529, 14038) is 1.

Step7

f) gcd( 11111, 111111)

By using  Euclidean algorithm

As 1 is the last nonzero remainder.

Therefore, gcd( 11111, 111111) is 1.

Step 2 of 3

Step 3 of 3

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Use the Euclidean algorithm to finda) gcd(1,