Answer :

Final answer:

There are 80 integers that are less than or equal to 195 and relatively prime to it based on Euler's totient function calculation. Hence, none of the options (A. 66 B. 65 C. 64 D. 63) provided in the question are correct.

Explanation:

To solve this problem, you need to understand the concept of relatively prime numbers and Euler's Totient Function. Two numbers are relatively prime if their greatest common divisor (GCD) is 1. Euler's totient function (φ(n)) gives the count of integers that are relatively prime to a given number 'n'.

195 can be factored into the prime numbers 3, 5, and 13. The formula for Euler's totient function is φ(n) = n * (1-1/p1) * (1-1/p2)...(1-1/pn), where p1, p2, pn are the distinct prime numbers that divide the given number. So, φ(195) = 195 * (1-1/3) * (1-1/5) * (1-1/13) = 1920/3 = 80.

, which means, there are 80 integers that are less than or equal to 195 and relatively prime to it. This implies that none of the four options given in the question are correct.

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