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Discrete Mathematics and its Applications

Kenneth H. Rosen

$Math Studyset Vaia Explanations$ Math

7th Edition · 808 pages

ISBN: 9780073383095

Chapter 4: Q41E (page 273)

Use the extended Euclidean algorithm to express gcd(26,91) as a linear combination of 26 and 91.

Short Answer

Expert verified

gcd(26,91) = 31

Linear combinations $(- 3) . 26 + 1.91 = 13$

Step by step solution

Step: 1

By using extended Euclidean algorithm

26 is not a prime, it is a composite number.

But 91 is a prime number it can be broken down to other prime factors.

Step: 2

gcd(26,91) = 31

Linear combinations $(- 3) . 26 + 1.91 = 13$

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Most popular questions from this chapter

Show that if aand d are positive integers, then there are integers and rsuch that a = dq + r where -d/2 < r < d/2 .

33. Show that a positive integer is divisible by 3 if and only if the difference of the sum of its binary digits in even numbered positions and the sum of its binary digits in odd-numbered positions is divisible by 3.

What are the greatest common divisors of these pairs of integers?

a) $2^{2} \cdot 3^{3} \cdot 5^{5}, 2^{5} \cdot 3^{3} \cdot 5^{2}$

b) $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13, 2^{11} \cdot 3^{9} \cdot 11 \cdot 17^{14}$

c) 17, $17^{17}$

d) $2^{2} \cdot 7, 5^{3} \cdot 13$

e) 0, 5

f) $2 \cdot 3 \cdot 5 \cdot 7, 2 \cdot 3 \cdot 5 \cdot 7$

39. Show that the integer m with one’s complement representation $(a_{n - 1} a_{n - 2} \dots a_{1} a_{0})$ can be found using the equation $m = - a_{n - 1} (2^{n - 1} - 1) + a_{n - 2} 2^{n - 2} + \dots . + a_{1} \cdot 2 + a_{0}$

Use exercise 36 to show that if a and b are positive integers, then $\gcd (2^{a} - 1, 2^{b} - 1) = 2^{\gcd (a, b)} - 1$

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Use the extended Euclidean algorithm to express gcd(26,91) as a linear combination of 26 and 91.

Short Answer

Step by step solution

Step: 1

Step: 2

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Most popular questions from this chapter

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