Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 14: Problem 16

Two integers are selected at random, say from the integers 1 through \(10^{9} .\) Calculate the probability of their being relatively prime. Hint: The probability of their not both being even is \(1-1 / 2^{2}\).

Short Answer

Expert verified
The probability that two integers are relatively prime is approximately 0.607927.

Step by step solution

01

- Understand 'Relatively Prime'

Two integers are said to be relatively prime if their greatest common divisor (GCD) is 1.
02

- Total Number of Pairs

The total number of pairs of integers from 1 to \(10^9\) is \((10^9) \times (10^9)\).
03

- Use Euler's Totient Function

Euler's Totient Function, \(\phi(n)\), gives the number of integers up to \(n\) that are relatively prime to \(n\). The probability that two integers are relatively prime is given by \(\frac{6}{\pi^2}\).
04

- Calculate the Probability

The probability that two random integers are relatively prime can be directly used as \(\frac{6}{\pi^2}\).
05

- Apply the Formula

The probability of two integers being relatively prime is \(\frac{6}{\pi^2} \approx 0.607927\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

relatively prime integers
Two integers are called relatively prime if they share no common factors other than 1.
If their greatest common divisor (GCD) is 1, they are relatively prime. For example, 8 and 15 are relatively prime because the only positive integer that divides both of them evenly is 1.
Checking if two numbers are relatively prime involves computing their GCD.
This concept is crucial in many areas of number theory and cryptography.
It helps us understand the structure and distribution of numbers.
Euler's Totient Function
Euler's Totient Function, denoted as \(\theta(n)\), counts the number of integers up to \(n\) that are relatively prime to \(n\).
For instance, \(\theta(9) = 6\) because the numbers 1, 2, 4, 5, 7, and 8 are relatively prime to 9.
It is a key tool in number theory and has various applications in fields like cryptography.
Euler's Totient Function can be used to generalize probabilities, helping to calculate the likelihood of numbers being relatively prime.
greatest common divisor (GCD)
The greatest common divisor (GCD) of two numbers is the largest number that divides both of them without leaving a remainder.
To determine if two integers are relatively prime, we need to find their GCD.
The Euclidean algorithm is a popular method for finding the GCD. For example, the GCD of 8 and 12 is 4 because 4 is the largest number that divides both 8 and 12 without a remainder.
If the GCD is 1, the numbers are relatively prime.
random selection of integers
When we talk about the random selection of integers, we refer to choosing numbers without any specific pattern or bias.
If we select two integers randomly from a large set, we can use probability theory to estimate their properties.
In the context of our problem, we calculate the likelihood that two randomly chosen integers are relatively prime. This probability is approximately \(0.607927\) or \(\frac{6}{\pi^2}\).
Understanding the concept of random selection helps us make sense of probabilistic results in number theory.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The \(n\) quantities \(x_{1}, x_{2}, \ldots, x_{n}\) are statistically independent; each has a Gaussian distribution with zero mean and a common variance: $$ \left\langle x_{i}\right\rangle=0 \quad\left\langle x_{i}^{2}\right\rangle=\sigma^{2} \quad\left\langle x_{i} x_{j}\right\rangle=0 $$ (a) \(n\) new quantities \(y_{1}\) are defined by \(y_{i}=\sum_{j} M_{i j} x_{j}\), where \(M\) is an orthogonal matrix. Show that the \(y_{i}\) have all the statistical properties of the \(x_{l}\); that is, they are independent Gaussian variables with $$ \left\langle y_{i}\right\rangle=0 \quad\left\langle y_{i}^{2}\right\rangle=\sigma^{2} \quad\left\langle y_{t} y_{j}\right\rangle=0 $$ (b) Choose \(y_{1}=\sqrt{\frac{1}{n}}\left(x_{1}+x_{2}+\cdots+x_{n}\right)\), and the remaining \(y_{1}\) arbitrarily, subject only to the restriction that the transformation be orthogonal. Show thereby that the mean \(\bar{x}=\frac{1}{n}\left(x_{1}+\cdots+x_{n}\right)\) and the quantity \(s=\sum_{i-1}^{n}\left(x_{i}-\bar{x}\right)^{2}\) are statistically independent random variables. What are the probability distributions of \(\bar{x}\) and \(s ?\) (c) One often wishes to test the hypothesis that the (unknown) mean of the Gaussian distribution is really zero, without assuming anything about the magnitude of \(\sigma .\) Intuition suggests \(\tau=\bar{x} / \sqrt{s}\) as a useful quantity. Show that the probability distribution of the random variable \(\tau\) is $$ p(\tau)=\sqrt{\frac{n}{\pi}} \frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)} \frac{1}{\left(1+n \tau^{2}\right)^{n / 2}} $$ The crucial feature of this distribution (essentially the so-called Student \(t\)-distribufion \(^{\top}\) ) is that it does not involve the unknown parameter \(\sigma\).

In the statistical theory of nucleat reactions. a typical cross section is assumed to be given by $$ \sigma(E)=\left|\sum_{i} \frac{\gamma_{i}}{E-E_{l}-i\left(\Gamma_{i} / 2\right)}\right|^{2} $$ where the sum runs over many resonances, each laving a resonance energy \(E_{i}\), partial width \(\gamma_{i}\), and total width \(\Gamma_{i}\), With the assumptions (1) all \(\Gamma_{i}\) are cqual to a (real) constant 5 (2) the \(\gamma /\) are indcpendent randomly distributed real numbers with $$ \left\langle\gamma_{i}\right\rangle=0 \quad\left\langle\gamma_{i}^{2}\right\rangle=x^{2}=\text { constant } $$ (3) the \(E_{i}\) are equally spaced, with spacing \(D\) (4) \(D \ll \Gamma\) evaluate \(\langle\sigma(E)\rangle,\left\langle\sigma^{2}(E)\right\rangle\), and \(\langle\sigma(E) \sigma(E+k)\rangle\) and show that $$ \frac{\langle\sigma(E)\rangle^{2}}{\left\langle\sigma^{2}(E)\right\rangle}=\frac{1}{2} \quad \frac{\left\langle[\sigma(E)-\sigma(E+k)]^{2}\right\rangle}{\left\langle\sigma^{2}(E)\right\rangle}=\frac{k^{2}}{k^{2}+\Gamma^{2}} $$

(a) Consider the function \(F(q)=(q-i a-\mu b)^{2}\), where a and \(\mathbf{b}\) are constant vectors, while for each choice of the vector \(q\) the scalars \(\hat{A}\) and \(\mu\) are adjusted so as to minimize \(F(q) .\) Show that this minimum \(F(\mathbf{q})=\mathbf{q}_{\perp}{ }^{2}\), where we define \(\mathbf{q}_{\|}\)and \(\mathbf{q}_{\perp}\) by \(\mathbf{q}=\mathbf{q}_{\|}+\mathbf{q}_{\perp}\), with \(\mathbf{q}_{\|}\)and \(\mathbf{q}_{1}\) parallel and perpendicular, respectively, to the plane containing a and \(\mathrm{b}\). (b) Suppose a variable \(y\) is known to be a linear function of \(x, y=\) \(\alpha x+\beta\), with \(x\) and \(\beta\) unknown constants. in order to determine these constants experimentally, \(y\) is measured at \(N\) different values of the variable \(x\), and a least squares fit of these data is made. If the experimental values of \(y\) have equal standard errors \(\sigma\), the fit \(i_{n}\) made by choosing \(a\) and \(b\) to minimize the quantity $$ \chi^{2}=\sum_{i=1}^{N} \frac{1}{\sigma^{2}}\left(y_{i}-a x_{i}-b\right)^{2} $$ so that \(a\) and \(b\) are lcast squares estimates of \(\alpha\) and \(\beta\). Show that the random variable \(\chi^{2}\) has the chi-square distribution ( \(14-73\) ) but with \(N-2\) degrees of freedom.

Measurements of the differential cross section for a nuclear reaction at several angles yield the following data; \(\begin{array}{cccccc}\theta & 30^{\circ} & 45^{3} & 90^{\circ} & 120^{\circ} & 150^{\circ} \\ \sigma(0) & 11 & 13 & 17 & 17 & 14 \\ \text { error } & \pm 1.5 & \pm 1.0 & \pm 2.0 & \pm 2.0 & \pm 1.5\end{array}\) The units of cross section are \(10^{-30} \mathrm{~cm}^{2} /\) steradian. (a) Make a least-squares fit to \(\sigma(\theta)\) of the form $$ \sigma(\theta)=A+B \cos \theta+C \cos ^{2} \theta $$ Give values and errors for \(A, B, C\). (b) Find the total cross section \(\sigma=\int \sigma(\theta) d \Omega\) and its error. (c) Find the differential cross section at \(0^{\circ}\) and its error.

The random variable \(x\) has the probability distribution $$ f(x)=e^{-x} \quad(0
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free