Question

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?

• A. 1
• B. 8
• C. 4
• D. 0
• E. 12

Hint:

Integers that are coprime with all others are usually prime numbers, as they have no divisors other than 1.

Answer 1

The Correct Option is B

Step 1: Understand the problem.
We need to find how many integers between 2 and 40 are coprime with every other number in that range.
Step 2: Find the numbers that are coprime with all others.
An integer is coprime with all other integers if its only common divisor with all other integers is 1. We need to find how many such numbers exist.
Step 3: Apply the conditions.
The answer is 8, since these integers are the prime numbers between 2 and 40.
Final Answer: \[ \boxed{8} \]