Final answer:
The Euler totient function, denoted as φ(n), counts the number of positive integers less than or equal to n that are relatively prime to n. To find the Euler totient of a number, we need to find the prime factorization of the number and use the formula φ(n) = n(1 - 1/p1)(1 - 1/p2)...(1 - 1/pn).
Step-by-step explanation:
The Euler totient function, denoted as φ(n), counts the number of positive integers less than or equal to n that are relatively prime to n. To find the Euler totient of a number, we need to find the prime factorization of the number and use the formula φ(n) = n(1 - 1/p1)(1 - 1/p2)...(1 - 1/pn), where p1, p2, ..., pn are the distinct prime factors of n.
a. For 23, as 23 is a prime number, there are 22 positive integers less than 23 that are relatively prime to it. Therefore, φ(23) = 22.
b. For 3139, we are given that 3139 = 43 * 73. Using the formula, we can calculate φ(3139) = 3139(1 - 1/43)(1 - 1/73) = 3139(42/43)(72/73) = 2856.
c. For 360, we are given that 360 = 8 * 9 * 5. Using the formula, we can calculate φ(360) = 360(1 - 1/2)(1 - 1/3)(1 - 1/5) = 360(1/2)(2/3)(4/5) = 96.