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Chapter 8: Problem 18

Compute \(\phi(n)\) for \(n\) equal to (a) 5186 ; (b) 5187 ; (c) 5188 .

Short Answer

Expert verified
\(\phi(5186)\) = 2520, \(\phi(5187)\) = 5186 and \(\phi(5188)\) = 1728.

Step by step solution

01

Factorization and Computation for 5186

First, factorize the number. 5186 can be written as 2 \times 37 \times 70. Therefore, \(\phi(5186)\) = \(5186 \times (1 - \frac{1}{2}) \times (1 - \frac{1}{37}) \times (1 - \frac{1}{70})\) = 2520.
02

Factorization and Computation for 5187

Next, factorize 5187. Since 5187 is a prime number, we use the property \(\phi(p)\) = \(p-1\) for a prime number \(p\), which gives \(\phi(5187)\) = \(5187-1\) = 5186.
03

Factorization and Computation for 5188

Lastly, factorize 5188. It can be written as 2 \times 2 \times 3 \times 433. So, \(\phi(5188)\) = \(5188 \times (1 - \frac{1}{2}) \times (1 - \frac{1}{3}) \times (1 - \frac{1}{433})\) = 1728

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