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 one year ago
```
How many positive integers less than or equal to 500 have exactly 3 divisors?
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How is my answer wrong?
 one year ago
``` How many positive integers less than or equal to 500 have exactly 3 divisors? ``` How is my answer wrong?

This Question is Closed

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.0\[3 = 3 \times 1 \]So the numbers we're looking for are in the form \(a^{3  1}b^{1 1} = a^2\)

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.0There are \(22\) perfect squares \(\le 500\). So my answer turns out to be \(22\)

LolWolf
 one year ago
Best ResponseYou've already chosen the best response.0Could it also include prime cases? In opposite of only non squarefree integers. Because, note that 30 is squarefree, yet has 3 divisors (that are not units or multiplied by units).

LolWolf
 one year ago
Best ResponseYou've already chosen the best response.0Well, sorry, by the "prime cases" I mean that they only have prime divisors.

RadEn
 one year ago
Best ResponseYou've already chosen the best response.2looks it is be a square number

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.0what should I do now?

LolWolf
 one year ago
Best ResponseYou've already chosen the best response.0Plus, I don't quite understand what the case is with them necessarily being square numbers? How'd you derive that?

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.0I just showed my work.

RadEn
 one year ago
Best ResponseYou've already chosen the best response.2a^2 always have (2+1) factors, in other words a^2 have exactly 3 divisors with a must be a prime number

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.030 has more than 3 divisors

LolWolf
 one year ago
Best ResponseYou've already chosen the best response.030 has how many, within these rules? Are you counting units and unit transformations?

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.0@RadEn But 21 is incorrect too!

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.030 has 1,2,3,6,10,30 as its divisors.

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.0Oh, a prime number? But why so?

LolWolf
 one year ago
Best ResponseYou've already chosen the best response.0Oh, okay, so we're counting improper divisors. Then, yes, the answer must be of the form: \(p^2\) for some prime \(p\).

RadEn
 one year ago
Best ResponseYou've already chosen the best response.2factors of 2^2 = {1,2,4} factors of 3^2 = {1,3,9} factors of 5^2 = {1,5,25} .... so on

LolWolf
 one year ago
Best ResponseYou've already chosen the best response.0Because, 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.

LolWolf
 one year ago
Best ResponseYou've already chosen the best response.0(Where \(p, q \ne 1\).

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.02,3,5,7,11,13,17,19 are the primes below 22. So should the answer be 8?

LolWolf
 one year ago
Best ResponseYou've already chosen the best response.0Also, don't forget numbers of the form: \[ n=pq \]Where \(p, q\) are prime.

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.0@LolWolf lol, that has 4 divisors

LolWolf
 one year ago
Best ResponseYou've already chosen the best response.0They have no more than three divisors. No, the answer is not, note that 6=2*3 also has 3 divisors.

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.06 has the divisors 1, 2, 3, 6

LolWolf
 one year ago
Best ResponseYou've already chosen the best response.0Oh, jeez, you're counting improper... BAH. Yes.

LolWolf
 one year ago
Best ResponseYou've already chosen the best response.0Then, yes, that's the case, it would be 8, indeed.

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.0Yes, it's 8. Thanks @RadEn!
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