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Chapter 4: Problem 43

Use the extended Euclidean algorithm to express \(\operatorname{gcd}(144,89)\) as a linear combination of 144 and \(89 .\)

Short Answer

Expert verified
1 = 34 \times 144 - 55 \times 89

Step by step solution

01

- Apply the Euclidean Algorithm

To find the gcd of 144 and 89 using the Euclidean algorithm, we perform the division steps until the remainder is 0.First, divide 144 by 89:\[144 = 89 \times 1 + 55\]Next, divide 89 by 55:\[89 = 55 \times 1 + 34\]Next, divide 55 by 34:\[55 = 34 \times 1 + 21\]Next, divide 34 by 21:\[34 = 21 \times 1 + 13\]Next, divide 21 by 13:\[21 = 13 \times 1 + 8\]Next, divide 13 by 8:\[13 = 8 \times 1 + 5\]Next, divide 8 by 5:\[8 = 5 \times 1 + 3\]Next, divide 5 by 3:\[5 = 3 \times 1 + 2\]Next, divide 3 by 2:\[3 = 2 \times 1 + 1\]Finally, divide 2 by 1:\[2 = 1 \times 2 + 0\]So, the greatest common divisor (gcd) is 1.
02

- Back Substitution

Now, use back substitution to express 1 as a linear combination of 144 and 89.Starting from:\[1 = 3 - 2 \times 1\]Substitute 2:\[2 = 5 - 3 \times 1\]So, we get:\[1 = 3 - (5 - 3 \times 1) = 3 \times 2 - 5\]Substitute 3:\[3 = 8 - 5 \times 1\]So, we get:\[1 = (8 - 5) \times 2 - 5 = 8 \times 2 - 5 \times 3\]Substitute 5:\[5 = 13 - 8\]So, we get:\[1 = 8 \times 2 - (13 - 8) \times 3 = 8 \times 5 - 13 \times 3\]Substitute 8:\[8 = 21 - 13\]So, we get:\[1 = (21 - 13) \times 5 - 13 \times 3 = 21 \times 5 - 13 \times 8\]Substitute 13:\[13 = 34 - 21\]So, we get:\[1 = 21 \times 5 - (34 - 21) \times 8 = 21 \times 13 - 34 \times 8\]Substitute 21:\[21 = 55 - 34\]So, we get:\[1 = (55 - 34) \times 13 - 34 \times 8 = 55 \times 13 - 34 \times 21\]Substitute 34:\[34 = 89 - 55\]So, we get:\[1 = 55 \times 13 - (89 - 55) \times 21 = 55 \times 34 - 89 \times 21\]Substitute 55:\[55 = 144 - 89\]So, we get:\[1 = (144 - 89) \times 34 - 89 \times 21 = 144 \times 34 - 89 \times 55\]
03

- Combine Terms

Combine like terms to express 1 as a linear combination of 144 and 89:\[1 = 144 \times 34 - 89 \times 55\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

gcd
The gcd, or greatest common divisor, of two integers is the largest integer that divides both numbers without leaving a remainder.
It is a fundamental concept in number theory and is often used in various mathematical computations.
Understanding gcd is essential for solving problems that involve divisibility and for finding common factors between numbers.
In this solution, it is crucial to determine the gcd of 144 and 89 to express it later as a linear combination.
linear combination
A linear combination of two numbers, say 144 and 89, means expressing their gcd using the form:
\[ ax + by = \text{gcd}(a, b) \]
Where \(a\) and \(b\) are coefficients.
The combination derives from back substituting remainders obtained through the Euclidean Algorithm.
Knowing how to form a linear combination is important for solving linear Diophantine equations and is a key concept in algebra.
back substitution
Back substitution is the process used to express the gcd as a linear combination of the original numbers.
This method starts from the smallest remainders obtained in the Euclidean algorithm and works its way back up.
The goal is to re-express each remainder until we are able to formulate the gcd in terms of the original numbers. Below are the steps from the solution:
- We start from 1 defined from the last non-zero remainder
- Replace intermediate remainders step-by-step
Eventually, you reach back to express the gcd using the original numbers.
Euclidean algorithm
The Euclidean algorithm is a method for finding the gcd of two integers.
It involves repeated division and taking remainders until the remainder is zero.
The last non-zero remainder is the gcd. Here鈥檚 the process applied in our exercise:
- Divide 144 by 89 to get a quotient and remainder
- Continue dividing the previous divisor by the remainder until the remainder is zero
It鈥檚 an efficient algorithm, foundational for understanding more complex arithmetic and number theory principles.

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Most popular questions from this chapter

In Exercises \(31-32\) suppose that Alice and Bob have these public keys and corresponding private keys: \(\left(n_{\text { Alice }}, e_{\text { Alice }}\right)=\) \((2867,7)=(61 \cdot 47,7), \quad d_{\text { Alice }}=1183\) and \(\left(n_{\text { Bob }}, e_{\text { Bob }}\right)=\) \((3127,21)=(59 \cdot 53,21), d_{\text { Bob }}=1149 .\) First express your answers without carrying out the calculations. Then, using a computational aid, if available, perform the calculation to get the numerical answers. Alice wants to send to all her friends, including Bob, the message "SELL EVERYTHING" so that he knows that she sent it. What should she send to her friends, assuming she signs the message using the RSA cryptosystem.

Describe the steps that Alice and Bob follow when they use the Diffie-Hellman key exchange protocol to generate a shared key. Assume that they use the prime \(p=101\) and take \(a=2,\) which is a primitive root of \(101,\) and that Alice selects \(k_{1}=7\) and \(\mathrm{Bob}\) selects \(k_{2}=9 .\) (You may want to use some computational aid.)

A parking lot has 31 visitor spaces, numbered from 0 to \(30 .\) Visitors are assigned parking spaces using the hashing function \(h(k)=k\) mod \(31,\) where \(k\) is the number formed from the first three digits on a visitor's license plate. a) Which spaces are assigned by the hashing function to cars that have these first three digits on their license plates: \(317,918,007,100,111,310 ?\) b) Describe a procedure visitors should follow to find a free parking space, when the space they are assigned is occupied. Another way to resolve collisions in hashing is to use double hashing. We use an initial hashing function \(h(k)=k \bmod p,\) where \(p\) is prime. We also use a second hashing function \(g(k)=(k+1) \bmod (p-2) .\) When a collision occurs, we use a probing sequence \(h(k, i)=(h(k)+i \cdot g(k)) \bmod p .\)

Find the first eight terms of the sequence of four-digit pseudorandom numbers generated by the middle square method starting with 2357.

Show that if \(p\) is an odd prime, then there are exactly \((p-1) / 2\) quadratic residues of \(p\) among the integers \(1,2, \ldots, p-1\) If \(p\) is an odd prime and \(a\) is an integer not divisible by \(p\) , the Legendre symbol \(\left(\frac{a}{p}\right)\) is defined to be 1 if \(a\) is a quadratic residue of \(p\) and \(-1\) otherwise.
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