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

Answer: (C)

468

Answer 2

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): $468$