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

 Jun 25, 2018
 #1
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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

 

 

cool cool cool

 Jun 25, 2018

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