Consider the Euler’s phi function given by \(\phi(n) = n \underset{p/n}{\Pi }…
2019
Consider the Euler’s phi function given by
\(\phi(n) = n \underset{p/n}{\Pi } \bigg( 1 – \frac{1}{p} \bigg)\)
where \(𝑝\) runs over all the primes dividing \(𝑛\). What is the value of \(𝜙(45)\)?
- A.
3
- B.
12
- C.
6
- D.
24
Attempted by 1 students.
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Correct answer: D
Solution:
Factorize the number: 45 = 3^2 × 5.
Use Euler’s phi formula:
φ(n) = n × ∏(1 − 1/p) over primes p dividing n.
Compute the product for the prime factors: (1 − 1/3) × (1 − 1/5) = (2/3) × (4/5) = 8/15.
Multiply by 45: φ(45) = 45 × 8/15 = 3 × 8 = 24.
Therefore φ(45) = 24.
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