Number System Question 5 - CAT 2023

Step-by-Step Explanation

• Analyze the prime factor structure:

The total number of factors is 15. The composite factorization of 15 can be split in two ways: 15 × 1 or 5 × 3.

• Case 1 (p¹⁴): The smallest number of this form is 2¹⁴ = 16384 (too large).
• Case 2 (p₁⁴ × p₂²): This form yields much smaller numbers. To minimize the values, use the smallest available prime bases (2,3,5).
• Evaluate the smallest combinations:
  • Smallest Value: Assign the larger exponent (4) to the smaller prime base (2), and the smaller exponent (2) to the next prime base (3):

N₁ = 2⁴ × 3² = 16 × 9 = 144

• Second Smallest Value: Swap the prime bases for the exponents:

N₂ = 2² × 3⁴ = 4 × 81 = 324

(Note: Using the next prime 5, like 2⁴ × 5² = 400*, gives a larger value).*

• Sum of the first two values = 144 + 324 = 468

Correct Answer: 468

💡 FundaMakers Tip:

When minimizing a number given a fixed number of factors, always assign the largest exponents to the smallest available prime bases. This approach keeps the numerical value as low as possible.