Arun selected an integer \( x \) between 2 and 40, both inclusive. He noticed that the greatest common divisor of the selected integer \( x \) and any other integer between 2 and 40, both inclusive, is 1.
How many different choices for such an \( x \) are possible?
Show Hint
Integers that are coprime with all others are usually prime numbers, as they have no divisors other than 1.
Step 1: Problem Definition. Determine the count of integers within the range [2, 40] that are relatively prime to all other integers in the same range. Step 2: Identify Coprime Numbers. A number is considered coprime with all others in a set if its greatest common divisor (GCD) with each of those other numbers is 1. We must count how many such numbers exist. Step 3: Solution Derivation. The count is 8. This is because the prime numbers in the range [2, 40] satisfy the condition. Final Answer: \[\boxed{8}\]