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

We know that the number of factors of two given numbers is \(15\).
The factors of \(15 =1, 3, 5\) and \(15\)
The number of factors of N \(=(p + 1) . (q + 1)\)
Where \(N\) is represented as \(a^p . b^q\), with \(a\) and \(b\) being prime numbers.
Thus, \(p+1 = 3\)
\(⇒ p = 2\)
And, \(q+1 = 5\)
\(⇒ q = 4\)
and the prime numbers \(a\) and \(b\) are \(2\) and \(3\).
Hence, the smallest value of \(N = 2^4 \times 3^2 = 16 \times 9= 144\)
and the second smallest value of \(N = 2^2\times 3^4 = 4 \times 81= 324\)
Now, the sum \(= 144+324 = 468\)

The correct  option is (C): \(468\)