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## ParthKohli 2 years ago  How many positive integers less than or equal to 500 have exactly 3 divisors?  How is my answer wrong?

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1. ParthKohli

$3 = 3 \times 1$So the numbers we're looking for are in the form $$a^{3 - 1}b^{1 -1} = a^2$$

2. ParthKohli

There are $$22$$ perfect squares $$\le 500$$. So my answer turns out to be $$22$$

3. LolWolf

Could it also include prime cases? In opposite of only non square-free integers. Because, note that 30 is square-free, yet has 3 divisors (that are not units or multiplied by units).

4. ParthKohli

Oh... not prime

5. LolWolf

Well, sorry, by the "prime cases" I mean that they only have prime divisors.

6. RadEn

looks it is be a square number

7. RadEn

except 1

8. ParthKohli

oh...

9. ParthKohli

what should I do now?

10. LolWolf

Plus, I don't quite understand what the case is with them necessarily being square numbers? How'd you derive that?

11. ParthKohli

I just showed my work.

12. RadEn

a^2 always have (2+1) factors, in other words a^2 have exactly 3 divisors with a must be a prime number

13. ParthKohli

30 has more than 3 divisors

14. LolWolf

30 has how many, within these rules? Are you counting units and unit transformations?

15. ParthKohli

@RadEn But 21 is incorrect too!

16. ParthKohli

30 has 1,2,3,6,10,30 as its divisors.

17. ParthKohli

and 5

18. ParthKohli

Oh, a prime number? But why so?

19. LolWolf

Oh, okay, so we're counting improper divisors. Then, yes, the answer must be of the form: $$p^2$$ for some prime $$p$$.

20. RadEn

factors of 2^2 = {1,2,4} factors of 3^2 = {1,3,9} factors of 5^2 = {1,5,25} .... so on

21. ParthKohli

oh.

22. LolWolf

Because, assume that $$a$$ is not prime, then: $a=pq$For some $$p, q\in \mathbb{Z}$$. So: $p|(pq)^2, p^2|(pq)^2, q|(pq)^2, pq|(pq)^2$Et al. Which is greater than 3 divisors. Hence, the number must be prime.

23. LolWolf

(Where $$p, q \ne 1$$.

24. ParthKohli

2,3,5,7,11,13,17,19 are the primes below 22. So should the answer be 8?

25. LolWolf

Also, don't forget numbers of the form: $n=pq$Where $$p, q$$ are prime.

26. ParthKohli

Oh Lord.

27. LolWolf

Jaja, yes.

28. RadEn

yes, the answer is 8

29. ParthKohli

@LolWolf lol, that has 4 divisors

30. LolWolf

They have no more than three divisors. No, the answer is not, note that 6=2*3 also has 3 divisors.

31. ParthKohli

6 has the divisors 1, 2, 3, 6

32. LolWolf

Oh, jeez, you're counting improper... BAH. Yes.

33. LolWolf

I forget.

34. LolWolf

Then, yes, that's the case, it would be 8, indeed.

35. ParthKohli

Yes, it's 8. Thanks @RadEn!

36. RadEn

you're welcome :)

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