Answer :
Final answer:
There are 120 numbers in the range 1 to 143 that are relatively prime with 143, which is found by using Euler's totient function giving φ(143) = (11 - 1)(13 - 1) = 120.
Explanation:
To find how many numbers in the range 1...143 are relatively prime with 143, we must understand that a number is relatively prime to 143 if it shares no common factors with 143 other than 1. The number 143 is the product of two primes, specifically 11 and 13, since 143 = 11 × 13. So, a number is relatively prime to 143 if it is not divisible by either 11 or 13.
We can use the concept of Euler's totient function (φ), which gives the number of integers that are relatively prime to a given number. In the case of 143, we can calculate φ(143) by using the formula for the totient function for a product of two distinct prime numbers p and q, which is φ(pq) = (p - 1)(q - 1). So, φ(143) = φ(11 × 13) = (11 - 1)(13 - 1) = 10 × 12 = 120.
Therefore, there are 120 numbers in the range from 1 to 143 that are relatively prime with 143, answer choice (d).
