The value of Euler’s totient function φ(77) is :

The correct answer is Option (3) → 60

\(\phi(n)\) is the number of integers from 1 to n that are coprime to n. For n with distinct prime factors p, q, ...:

    \( \displaystyle \phi(n) = n\left(1-\frac{1}{p}\right)\left(1-\frac{1}{q}\right)\cdots \)

Here \(77 = 7 \times 11\):

    \( \displaystyle \phi(77) = 77\left(1-\frac{1}{7}\right)\left(1-\frac{1}{11}\right) \)

    \( \displaystyle = 77\times\frac{6}{7}\times\frac{10}{11} = 6\times 10 = 60 \)

Check by counting: of 1 to 77, the multiples of 7 (11 numbers) and of 11 (7 numbers) share only 77, so 11 + 7 − 1 = 17 numbers are not coprime, and 77 − 17 = 60.