Question

The sum of the first two natural numbers, each having 15 factors (including 1 and the number itself), is

• A. 348
• B. 412
• C. 468
• D. None of Above

Answer 1

The Correct Option is C

Finding Numbers with 15 Factors 

We are given that the number of factors of a number \( N \) is 15.

The number 15 has the following pairs of factor counts: \( 15 = 1 \times 15 = 3 \times 5 \).
Hence, possible exponent combinations for the prime factorization of \( N \) are:

• \( (p+1)(q+1) = 3 \times 5 \Rightarrow p = 2, q = 4 \)
• Here, \( N = a^2 \cdot b^4 \) or \( a^4 \cdot b^2 \), where \( a \) and \( b \) are distinct primes.

 

Case 1: \( N = 2^4 \times 3^2 \)

\( 2^4 = 16 \), \( 3^2 = 9 \) 
\( \Rightarrow N = 16 \times 9 = 144 \)

Case 2: \( N = 2^2 \times 3^4 \)

\( 2^2 = 4 \), \( 3^4 = 81 \) 
\( \Rightarrow N = 4 \times 81 = 324 \)

Sum of the Two Smallest Such Numbers:

\( 144 + 324 = 468 \)

Correct Answer: (C): \( \boxed{468} \)