Which integers have exactly 3 distinct positive factors?

[Isaak DeMaio]

1. Which integers have exactly 3 distinct positive factors?

Homework Equations

3. I would attempt this if I had any idea of what it meant. Can someone show me how to find one answer then I will find the other 2.

 

[Dick]

9 has three distinct positive factors, 1, 3 and 9, yes? What other numbers might have the same property?

 

[Isaak DeMaio]

9 has three distinct positive factors, 1, 3 and 9, yes? What other numbers might have the same property?

that's what I thought.
So 4... 1,2,4

25.. 1 5 25

wow I'm dumb.

 

[dextercioby]

So ? What's the answer then ?

 

[Isaak DeMaio]

4,9,25?

 

[Dick]

4,9,25?

There are more than that. How can you describe them (other than just having three factors)?

 

[Isaak DeMaio]

Would it just be all odd square numbers, including 4?

 

[Dick]

Would it just be all odd square numbers, including 4?

Not quite. 81 is an odd square. It doesn't work.

 

[HallsofIvy]

Square numbers, yes, but not all square numbers.

 

[Robert1986]

Would it just be all odd square numbers, including 4?

Close, but look at these numbers:

25 = 5^2, 9 = 3^2 both fit your description, however,
16 = 4^2, 81 = 9^2 do not.

Do you see thie difference? That is what I would try to do.

 

[Isaak DeMaio]

Square numbers, yes, but not all square numbers.

well 4,9,25,49

2^2, 3^2, 5^2, 7^2, 11^2, 13^2

81 is a perfect cube too.

So would it be all odd square numbers, that are not also perfect cubes...Including 4.

 

[Isaak DeMaio]

Close, but look at these numbers:

25 = 5^2, 9 = 3^2 both fit your description, however,
16 = 4^2, 81 = 9^2 do not.

Do you see thie difference? That is what I would try to do.

4 has three distinct factors, 1,2,4.
Good one though.

 

[dextercioby]

Only the prime numbers. These are less than odd numbers.

 

[Isaak DeMaio]

Only the prime numbers. These are less than odd numbers.

The question is "Which integers have exactly 3 distinct positive factors."
Prime number only have 2 factors, one and itself.

 

[Char. Limit]

He means the squares of the prime numbers.

 

[Isaak DeMaio]

He means the squares of the prime numbers.

Easier if he could say that in a full sentence.

 

[Char. Limit]

Easier if he could say that in a full sentence.

That's why I clarified for him.

 

[Isaak DeMaio]

That's why I clarified for him.

Gold Star.

 

[Robert1986]

4 has three distinct factors, 1,2,4.
Good one though.

Exactly, and 4 = 2^2.

The pattern that I was trying to get you to recognize was that 9 and 25 were squares of primes whereas 16 and 81 were squares of composites (as has since been pointed out.) 4, being a square of a prime - 2 - fits the description of the numbers you were searching for.

 

[dextercioby]

Gold Star.

Alright, so this problem you got solved with the help of the Physicsforums. As a further exercise, find the answer to this problem:

<Which integers have exactly 4 distinct positive factors ?>