Question

Let \( a_1 < a_2< \dots <a_n \) be the list of all prime numbers less than 25. Define \( X_i = \frac{b_i}{a_i} \), where \( b_i \) is the sum of all \( a_k \) where \( k \) ranges from 1 to \( n \), \( k \neq i \). Let \( B \) be the set of all integer-valued \( X_i \).
What is the SMALLEST element of \( B \)?

• A. 19
• B. 1
• C. 23
• D. 11
• E. 17

Hint:

To find the smallest integer in a set of fractional values, look for the highest denominator.

Answer 1

The Correct Option is B

Step 1: List of prime numbers less than 25.
The prime numbers less than 25 are: \[ a_1 = 2, a_2 = 3, a_3 = 5, a_4 = 7, a_5 = 11, a_6 = 13, a_7 = 17, a_8 = 19, a_9 = 23 \]
Step 2: Calculate the sum of all \( a_k \).
The sum of all prime numbers less than 25 is: \[ \text{Sum} = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 = 100 \]
Step 3: Calculate \( X_i \) for each prime number \( a_i \).
For each \( i \), we define \( X_i = \frac{b_i}{a_i} \), where \( b_i \) is the sum of all \( a_k \) except \( a_i \). Therefore, \[ b_i = 100 - a_i \] So, \[ X_1 = \frac{100 - 2}{2} = \frac{98}{2} = 49, \quad X_2 = \frac{100 - 3}{3} = \frac{97}{3} = 32.33, \quad \text{and so on.} \]
Step 4: Conclusion.
The smallest value of \( X_i \) is 1. Therefore, the smallest element of \( B \) is (B).