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

To solve this problem, we need to identify the integers \( x \) that are coprime with all the integers between 2 and 40 inclusive. This means we are looking for integers \( x \) such that the greatest common divisor (GCD) of \( x \) and any integer \( n \) from 2 to 40 is 1. Such integers \( x \) must themselves be coprime to 40. 

First, let's identify the factors of 40:

• Prime factors of 40 are: 2 and 5.

An integer \( x \) is coprime to 40 if it doesn't share any prime factors with 40. Therefore, \( x \) must not be divisible by 2 or 5. We now find such integers in the range 2 to 40.

Let's list the relevant candidates and check for divisibility:

• The numbers that are not divisible by 2 or 5 are: 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, and 39.

All these numbers listed are not divisible by 2 or 5, thus coprime to 40. Now, let's count these numbers: 3, 7, 9, 11, 13, 17, 19, 23, 27, 29, 31, 33, 37, and 39.

List: 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39

With these numbers confirmed, let's count how many integers are there:

• 3, 11, 13, 17, 19, 23, 29, 31, 37.

Therefore, there are 8 such numbers. The correct option is:

Result

8

Answer 2

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: \[ \boxed{8} \]