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Chapter 8: Problem 23

Find three values for \(n \in \mathbf{Z}^{+}\)where \(\phi(n)=16\).

Short Answer

Expert verified
The three values for \(n\) such that \(\phi(n) = 16\) are n = 17 and n = 32.

Step by step solution

01

Understanding Euler's Totient Function

Euler's totient function \(\phi(n)\) counts the positive integers up to a given integer \(n\) that are relatively prime to \(n\). Two numbers are called relatively prime, mutually prime, or co-prime if the only positive integer that divides both of them is 1.
02

Using Properties of Euler's Totient Function

It is known that for any prime number \(p\), \(\phi(p^k) = p^k - p^{k-1}\). This means that the possible values of \(\phi(n)\) are decreasing when moving from \(p^k\) to \(p^{k+1}\) where \(p\) is a prime number.
03

Finding values of n

We look for values of \(n\) such that \(\phi(n) = 16\). Do this by trying different primes and their powers until you get a result equal to 16. Using \(p=2\) we get \(\phi(2^5) = 2^5 - 2^4 = 32 - 16 = 16\), so \(n=32\) is one solution. Next, let's try \(p=17\) and we get \(\phi(17^1) = 17^1 - 17^0 = 16\), so \(n=17\) is another solution. Using p=3 isn't going to work because \(\phi(3^3) = 18 \neq 16\). Similarly for \(p=5\), because \(\phi(5^2) = 20 \neq 16\). So there aren't any more solutions.

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Most popular questions from this chapter

In how many ways can three \(x\) 's, three \(y\) 's, and three \(z\) 's be arranged so that no consecutive triple of the same letter appears?

In how many ways can Mrs. Ford distribute ten distinct books to her ten children (one book to each child) and then collect and redistribute the books so that each child has the opportunity to peruse two different books?

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a) When \(n\) balls, numbered \(1,2,3, \ldots, n\), are taken in succession from a container, a rencontre occurs if the \(m\) th ball withdrawn is numbered \(m\), for some \(1 \leq m \leq n\). Find the probability of getting (i) no rencontres; (ii) exactly one rencontre; (iii) at least one rencontre; and, (iv) \(r\) rencontres, where \(1 \leq r \leq n\). b) Approximate the answers to the questions in part (a).

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