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Find the smallest square that has at least $3$ different prime factors.

I tried finding the LCM of the $3$ smallest primes, $2$, $3$ and $5$, which is $30$, and $30^2$ equals $900$. But is this the right answer, and if yes, is there an easy way to work out this problem?

(I'm only a Year 7 (age $11-12$ for any non-UK people out there), so please explain clearly how you found the solution and if you have a formula for working out this problem, please make it so I can understand it.)

lioness99a
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bio
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  • Should the prime factors be all different? otherwise 16 is an answer – LM2357 Mar 22 '17 at 08:31
  • @user35508 yes they should I'll put that in the question – bio Mar 22 '17 at 08:31
  • Well, the only "gotcha" is that they didn't say 3 distinct prime factors. If the primes are required to be distinct, then your answer is correct. If they're not required to be distinct, then the answer is ... – quasi Mar 22 '17 at 08:32
  • @bio....but $8$ is not a perfect square – LM2357 Mar 22 '17 at 08:36
  • yes fair point I deleted the comment – bio Mar 22 '17 at 08:37
  • @bio the square is the smallest when the number itself is smallest. $30=235$ giving $900$ is the right answer. The next smallest is $42=237$. But the next one is not $66=2311$ – Lozenges Mar 22 '17 at 09:09
  • @bio: I think you made a very good argument. The easy way to work out that problem is exactly what you did. Now of course depending on the context of the question (I assume it came up somewhere in school),a teacher might like to hear something about unique prime factorization in the answer (if you don't know what that is, basically it means that when you calculate the product of prime numbers, you can't end up with a number that is divisible by some prime that you didn't put into the multiplication). – Ingix Mar 22 '17 at 09:12
  • @bio the smallest squares are ${900,1764,3600,4356,4900,\text{...}}$ – Lozenges Mar 22 '17 at 11:11

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Yes, that's the right answer, and it's also the best way of working it out in my opinion.

Squaring a number doesn't change which primes are factors, and a smaller (positive) number will have a smaller square, so you need to find the smallest number which has three different prime factors, then square it. For any given three primes, the smallest number which has those primes as factors is their LCM, which (because they are prime, and different) is just their product. So choosing smaller primes will give you a smaller answer, and therefore the best thing to do is take the three smallest primes, to get $(2\times3\times5)^2$. If the question had said "four distinct prime factors", it would be $(2\times3\times5\times7)^2$, and so on. (Similarly, if it had asked for the smallest cube with three distinct prime factors, the answer would be $2\times3\times5)^3$.)

Especially Lime
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