hi. can anybody explain me this question?

The decomposition of a number into prime factors will be of the form 2^{a}3^{b}5^{c} and so on. We also know that the number of possible factors is (a + 1)(b + 1)(c + 1).... Note that a, b and so on must be positive.

The question tells that this is 20. We need to find out whether

- this is possible with three distinct prime divisors. In other words, is it possible that (a + 1)(b + 1)(c + 1) = 20? For 3 indeed, as 2 \times 2 \times 5 = 20 to give an example.
- this is possible with four distinct prime divisors. Nope, and we can use the above example to explain. 20 = 2 \times 2 \times 5, and to have four distinct primes, we must be able to decompose 20 such that we get four primes. But we get only 3, and hence this isn’t possible.
- the case with five distinct prime divisors follows naturally.