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Chapter 4: Problem 16

Determine whether the integers in each of these sets are pairwise relatively prime. $$\begin{array}{ll}{\text { a) } 21,34,55} & {\text { b) } 14,17,85} \\\ {\text { c) } 25,41,49,64} & {\text { d) } 17,18,19,23}\end{array}$$

Short Answer

Expert verified
Pairwise relatively prime sets: a), c), d). Not pairwise relatively prime set: b).

Step by step solution

01

- Understanding Pairwise Relatively Prime

Pairwise relatively prime integers are a set of integers where every pair of integers within the set is relatively prime. This means that the greatest common divisor (GCD) of each pair is 1.
02

- Check Set a) 21, 34, 55

Calculate the GCD of each pair of numbers 1. GCD(21, 34): - Prime factors of 21: 3, 7 - Prime factors of 34: 2, 17 - Common factors: None - GCD: 1 2. GCD(21, 55): - Prime factors of 21: 3, 7 - Prime factors of 55: 5, 11 - Common factors: None - GCD: 1 3. GCD(34, 55): - Prime factors of 34: 2, 17 - Prime factors of 55: 5, 11 - Common factors: None - GCD: 1 Since all pairs have a GCD of 1, the integers in set a) are pairwise relatively prime.
03

- Check Set b) 14, 17, 85

Calculate the GCD of each pair of numbers 1. GCD(14, 17): - Prime factors of 14: 2, 7 - Prime factors of 17: 17 - Common factors: None - GCD: 1 2. GCD(14, 85): - Prime factors of 14: 2, 7 - Prime factors of 85: 5, 17 - Common factors: None - GCD: 1 3. GCD(17, 85): - Prime factors of 17: 17 - Prime factors of 85: 5, 17 - Common factors: 17 - GCD: 17 Since not all pairs have a GCD of 1, the integers in set b) are not pairwise relatively prime.
04

- Check Set c) 25, 41, 49, 64

Calculate the GCD of each pair of numbers 1. GCD(25, 41): - Prime factors of 25: 5 - Prime factors of 41: 41 - Common factors: None - GCD: 1 2. GCD(25, 49): - Prime factors of 25: 5 - Prime factors of 49: 7 - Common factors: None - GCD: 1 3. GCD(25, 64): - Prime factors of 25: 5 - Prime factors of 64: 2 - Common factors: None - GCD: 1 4. GCD(41, 49): - Prime factors of 41: 41 - Prime factors of 49: 7 - Common factors: None - GCD: 1 5. GCD(41, 64): - Prime factors of 41: 41 - Prime factors of 64: 2 - Common factors: None - GCD: 1 6. GCD(49, 64): - Prime factors of 49: 7 - Prime factors of 64: 2 - Common factors: None - GCD: 1 Since all pairs have a GCD of 1, the integers in set c) are pairwise relatively prime.
05

- Check Set d) 17, 18, 19, 23

Calculate the GCD of each pair of numbers 1. GCD(17, 18): - Prime factors of 17: 17 - Prime factors of 18: 2, 3 - Common factors: None - GCD: 1 2. GCD(17, 19): - Prime factors of 17: 17 - Prime factors of 19: 19 - Common factors: None - GCD: 1 3. GCD(17, 23): - Prime factors of 17: 17 - Prime factors of 23: 23 - Common factors: None - GCD: 1 4. GCD(18, 19): - Prime factors of 18: 2, 3 - Prime factors of 19: 19 - Common factors: None - GCD: 1 5. GCD(18, 23): - Prime factors of 18: 2, 3 - Prime factors of 23: 23 - Common factors: None - GCD: 1 6. GCD(19, 23): - Prime factors of 19: 19 - Prime factors of 23: 23 - Common factors: None - GCD: 1 Since all pairs have a GCD of 1, the integers in set d) are pairwise relatively prime.

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Key Concepts

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

Greatest Common Divisor (GCD)
The Greatest Common Divisor (GCD) of two or more integers is the largest integer that divides each of them without leaving a remainder. In simpler terms, it's the biggest number that can evenly split the given numbers.
For example, to find the GCD of 8 and 12:
- The factors of 8 are 1, 2, 4, and 8.
- The factors of 12 are 1, 2, 3, 4, 6, and 12.
- The common factors of both are 1, 2, and 4.
- Thus, the GCD is 4, as it's the largest of the common factors.

We often use GCD in problems involving pairwise relatively prime integers, where we need to check if sets of numbers share any factors other than 1. If the GCD of each pair in a set is 1, the numbers are pairwise relatively prime.
Prime Factors
Prime factors are the prime numbers that multiply together to give the original number. A prime number is a number greater than 1 that has no divisors other than 1 and itself.

To find prime factors:
- Break down the number into its smallest prime components.
For example, to find the prime factors of 18:
- 18 can be divided by 2 to give 9 (so, 2 is one prime factor).
- 9 can be divided by 3 to give 3 (so, 3 is another prime factor).
- Finally, 3 is already a prime number.
- Therefore, the prime factors of 18 are 2 and 3.

When checking for pairwise relatively prime integers, it's helpful to break each number down into its prime factors to see if they share any common prime factors. If they do, their GCD will be greater than 1.
Number Theory
Number theory is a branch of mathematics concerned with the properties and relationships of numbers, particularly integers. It plays a crucial role in various topics like prime numbers, divisibility, and the relationships between numbers.

Here are some key concepts within number theory useful for understanding the exercise:
- **Relatively Prime**: Two numbers are relatively prime if their GCD is 1. They don't share any prime factors.
- **Pairwise Relatively Prime**: A set of numbers is pairwise relatively prime if every pair of numbers within the set is relatively prime. This means the GCD of each pair in the set is 1.

In the exercise, we use number theory techniques to check for pairwise relatively prime integers. By calculating the GCD of each pair of numbers and examining their prime factors, we establish whether or not the sets of numbers meet the criteria for being pairwise relatively prime.

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Most popular questions from this chapter

Show that the integer \(m\) with one's complement representation \(\left(a_{n-1} a_{n-2} \ldots a_{1} a_{0}\right)\) can be found using the equation \(m=-a_{n-1}\left(2^{n-1}-1\right)+a_{n-2} 2^{n-2}+\cdots+a_{1} \cdot 2+a_{0}\).

Prove that the set of positive rational numbers is countable by showing that the function \(K\) is a one-to- one correspondence between the set of positive rational numbers and the set of positive integers if \(K(m / n)=p_{1}^{2 a_{1}} p_{2}^{2 a_{2}} \cdots \cdots p_{s}^{2 a_{s}} q_{1}^{2 b_{1}-1} q_{2}^{2 b_{2}-1} \ldots \cdots q_{t}^{2 b_{t}-1}\) where gcd \((m, n)=1\) and the prime-power factorizations of \(m\) and \(n\) are \(m=p_{1}^{a_{1}} p_{2}^{a_{2}} \cdots \cdot p_{s}^{a_{s}}\) and \(n=q_{1}^{b_{1}} q_{2}^{b_{2}} \cdots q_{t}^{b_{t}}\)

How many divisions are required to find gcd(34, 55) using the Euclidean algorithm?

How is the one's complement representation of the difference of two integers obtained from the one's complement representations of these integers?

The Vigen猫re cipher is a block cipher, with a key that is a string of letters with numerical equivalents \(k_{1} k_{2} \ldots k_{m},\) where \(k_{i} \in \mathbf{Z}_{26}\) for \(i=1,2, \ldots, m .\) Suppose that the numerical equivalents of the letters of a plaintext block are \(p_{1} p_{2} \ldots p_{m} .\) The corresponding numerical ciphertext block is \(\left(p_{1}+k_{1}\right)\) mod 26 \(\left(p_{2}+k_{2}\right) \bmod 26 \ldots\left(p_{m}+k_{m}\right)\) mod \(26 .\) Finally, we translate back to letters. For example, suppose that the key string is RED, with numerical equivalents \(1743 .\) Then, the plaintext ORANGE, with numerical equivalents \(141700130604,\) is encrypted by first splitting it into two blocks 141700 and 13 \(0604 .\) Then, in each block we shift the first letter by 17 , the second by \(4,\) and the third by \(3 .\) We obtain 52103 and 0410 \(07 .\) The cipherext is FVDEKH. Use the Vigen猫re cipher with key BLUE to encrypt the message SNOWFALL.
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