ParthKohli
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How many positive integers less than or equal to 500 have exactly 3 divisors?
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How is my answer wrong?
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ParthKohli
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\[3 = 3
\times 1 \]So the numbers we're looking for are in the form \(a^{3 - 1}b^{1 -1} = a^2\)
ParthKohli
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There are \(22\) perfect squares \(\le 500\). So my answer turns out to be \(22\)
LolWolf
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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).
ParthKohli
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Oh... not prime
LolWolf
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Well, sorry, by the "prime cases" I mean that they only have prime divisors.
RadEn
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looks it is be a square number
RadEn
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except 1
ParthKohli
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oh...
ParthKohli
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what should I do now?
LolWolf
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Plus, I don't quite understand what the case is with them necessarily being square numbers? How'd you derive that?
ParthKohli
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I just showed my work.
RadEn
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a^2 always have (2+1) factors, in other words a^2 have exactly 3 divisors
with a must be a prime number
ParthKohli
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30 has more than 3 divisors
LolWolf
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30 has how many, within these rules? Are you counting units and unit transformations?
ParthKohli
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@RadEn But 21 is incorrect too!
ParthKohli
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30 has 1,2,3,6,10,30 as its divisors.
ParthKohli
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and 5
ParthKohli
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Oh, a prime number? But why so?
LolWolf
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Oh, okay, so we're counting improper divisors. Then, yes, the answer must be of the form: \(p^2\) for some prime \(p\).
RadEn
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factors of 2^2 = {1,2,4}
factors of 3^2 = {1,3,9}
factors of 5^2 = {1,5,25}
.... so on
ParthKohli
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oh.
LolWolf
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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.
LolWolf
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(Where \(p, q \ne 1\).
ParthKohli
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2,3,5,7,11,13,17,19 are the primes below 22. So should the answer be 8?
LolWolf
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Also, don't forget numbers of the form:
\[
n=pq
\]Where \(p, q\) are prime.
ParthKohli
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Oh Lord.
LolWolf
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Jaja, yes.
RadEn
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yes, the answer is 8
ParthKohli
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@LolWolf lol, that has 4 divisors
LolWolf
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They have no more than three divisors.
No, the answer is not, note that 6=2*3 also has 3 divisors.
ParthKohli
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6 has the divisors 1, 2, 3, 6
LolWolf
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Oh, jeez, you're counting improper... BAH. Yes.
LolWolf
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I forget.
LolWolf
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Then, yes, that's the case, it would be 8, indeed.
ParthKohli
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Yes, it's 8.
Thanks @RadEn!
RadEn
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you're welcome :)