Bunuel wrote:

The function f(a) is defined for all positive integers a as the number of positive integers which are less than a and have no common factor with a other than 1. What is f(77)?

A. 60

B. 63

C. 66

D. 70

E. 76

We are given that the function f(a) is defined for all positive integers a as the number of positive integers that are less than a and have no common factor with a other than 1, and we need to determine f(77).

We can prime factor 77 as 7 x 11. So now we can express the question as: How many positive integers less than 77 do not have 7 or 11 as factors? Our first step is to determine the number of multiples of 7 and 11 that are less than 77.

For 11: 11, 22, 33, 44, 55, 66 = 6 multiples

For 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70 = 10 multiples

We can see that there are 6 + 10 = 16 integers that have common factors (other than 1) with 77. Since there are 76 positive integers less than 77 and 16 of them have common factors (other than 1) with 77, there are 76 - 16 = 60 numbers that have no common factors (other than 1) with 77.

Answer: A

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