Noting that 666=2x3^2x37, find phi(666) using Euler Theorem.
phi(666) =
2x3^2x37 1/(2x3x37) x (2-1)x (3-1)x(37-1) =
3 x 2 x 36 = 216
To find the value of φ(666) using Euler's theorem, we first need to understand what φ(n) represents. φ(n) is Euler's totient function, which gives the count of positive integers less than or equal to n that are coprime (relatively prime) to n. In other words, φ(n) denotes the number of positive integers k such that 1 ≤ k ≤ n and gcd(k, n) = 1.
To apply Euler's theorem, we need to express 666 as a product of prime factors. As noted in the question, 666 can be factorized as 2 × 3^2 × 37.
Euler's theorem states that if n is a positive integer and a is any integer coprime to n, then a^(φ(n)) ≡ 1 (mod n).
Now, let's find φ(666):
φ(666) = φ(2) × φ(3^2) × φ(37)
Since 2 is prime, φ(2) = 2 - 1 = 1
For φ(3^2), we know that if p is a prime number, then φ(p^k) = p^k - p^(k-1). Therefore, φ(3^2) = 3^2 - 3^(2-1) = 9 - 3 = 6
Similarly, as 37 is a prime number, φ(37) = 37 - 1 = 36
Now, we can calculate φ(666):
φ(666) = φ(2) × φ(3^2) × φ(37)
= 1 × 6 × 36
= 216
So, φ(666) = 216.