Not sure how to start this one. Any idea how to approach it?

Question

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

T Ronak · Accepted Answer

To solve this, first understand how to find the number of factors of a number.
If a number N = (a^p) &times; (b^q), where a and b are prime numbers, then the number of factors is (p+1)(q+1). Now we&rsquo;re told the number of factors is 15.
So we need (p+1)(q+1) = 15. The factor pairs of 15 are (3,5) and (5,3). That means (p, q) can be (2, 4) or (4, 2).
Now use smallest primes: a = 2, b = 3.
Try both:

2^4 &times; 3^2 = 144
  2^2 &times; 3^4 = 324

These are the smallest two numbers with 15 factors. Add them: 144 + 324 = 468

V Gupta · Answer

Was stuck here too, but once the method is clear, it&rsquo;s actually doable.

Use formula: if N = (a^p) &times; (b^q), number of factors = (p+1)(q+1)
Given: total factors = 15 &rarr; so (p+1)(q+1) = 15
Possible (p, q) pairs: (2,4) and (4,2)
Use smallest primes: a = 2 and b = 3
First number: 2^4 &times; 3^2 = 144
Second number: 2^2 &times; 3^4 = 324
Sum = 144 + 324 = 468

Answer is 468. Just needed to try both exponent combinations using smallest prime bases.