How many natural numbers less than 1000 have exactly three distinct positive integer divisors?

Guest Jun 25, 2018

#1**+2 **

This seems tougher than it really is

Note....that every * non-square *natural number will have an even number of divisors

For instance.....the divisors of 8 are 1 2 4 8

However....the perfect squares always have an odd number of divisors

For instance....the divsors of 16 are 1 2 4 8 and 16

And the perfect squares which have * prime *square roots will only have 3 divisors

For instance...9 has a square root of 3 which is prime....so...it will only have 3 divisors ⇒ 1, 3 and 9

So...the squares < 1000 which have prime square roots are

4 289

9 361

25 529

49 841

121 961

169

So...11 natural numbers < 1000 have only 3 divisors

CPhill Jun 25, 2018