The sum of the first two natural numbers, each having 15 factors (including 1 and the number itself), is
- 348
- 412
- 468
- None of Above
The Correct Option is C
Solution and Explanation
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): \( \boxed{468} \)
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