The difference between the largest and the smallest prime numbers below 100 is a multiple of two prime numbers. What is the sum of these prime numbers?

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?

Answer 1

The correct answer is option (C):

24

Here's how to solve this problem:

1. Identify the Prime Numbers: Recall that a prime number is a whole number greater than 1 that is only divisible by 1 and itself. Find the largest and smallest prime numbers less than 100. The smallest is 2, and the largest is 97.

2. Calculate the Difference: Subtract the smallest prime (2) from the largest prime (97): 97 - 2 = 95.

3. Prime Factorization: Determine the prime factors of the difference (95). We can divide 95 by 5 to get 19. Both 5 and 19 are prime numbers. Therefore, the prime factors of 95 are 5 and 19.

4. Calculate the Sum: Add the two prime factors together: 5 + 19 = 24.

Therefore, the sum of the two prime numbers is 24.

The difference between the largest and the smallest prime numbers below 100 is a multiple of two prime numbers. What is the sum of these prime numbers?

The Correct Option is C

Solution and Explanation

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