Question

A certain positive integer has exactly 20 positive divisors.

What is the largest number of primes that could divide the integer?

thx for the help in advance

Answer 1

9514 1404 393

Answer:

3

Explanation:

20 has at most 3 proper factors greater than 1: 2×2×5. Each of these can represent a prime factor of the number of interest, and is 1 more than that prime's power. That is, the number of interest (n) will have at most 3 prime factors p, q, r, and will be ...

n = p·q·r^4

_____

For some prime factorization ...

`\displaystyle n=\prod_(k=1)^m{p_k^(q_k)}`

The total number of divisors of n is ...

`\displaystyle\prod_(k=1)^m{(q_k+1)}`