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Today morning, I found some numbers which have 2 factor pairs accepting the condition that the largest of them divided by the other must be lower than the 2. For example, I found $720$ which satisfies the condition since their two factor pairs $30, 24$ and $36, 20$ and 36/20, 30/24 are smaller than two. Can you find smallest natural that does so? I found 180 at least. P.S. don't include square numbers.

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    $\begingroup$ Seems simple program will do this, as you do not need any mathematical investigation. $\endgroup$
    – zkutch
    Aug 25 '20 at 8:20
  • $\begingroup$ dividing gives the quotient , not the difference. $120=8\cdot 15=10\cdot 12$ is a smaller example, I have not checked yet whether it is the smallest. $\endgroup$
    – Peter
    Aug 25 '20 at 8:34
  • $\begingroup$ why not to take $60$? $\endgroup$
    – zkutch
    Aug 25 '20 at 8:36
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    $\begingroup$ You are looking for a number $n$ with two factors $x$ where $\sqrt{n} < x < \sqrt{2n}$. I think $n=120$ is the smallest with $10.95 < x <15.49$ so $12$ and $15$. With $60$ you have $7.75 < x <10.95$ but neither $8$ nor $9$ are factors of $60$ $\endgroup$
    – Henry
    Aug 25 '20 at 8:37
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    $\begingroup$ @zkutch you got the point! That's what I meant. $\endgroup$
    – user
    Aug 25 '20 at 8:47
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Here the PARI/GP code and the table (the second column means the number of possible pairs) :

gp > for(n=1,1000,v=select(m->(n/m^2<2)*(m^2<n),divisors(n));if(length(v)>=2,print(n,"   ",length(v))))
120   2
180   2
240   2
252   2
336   2
360   2
378   2
420   2
432   2
480   2
504   2
540   2
594   2
600   2
630   2
660   2
672   2
700   2
720   2
756   2
780   2
792   2
840   3
858   2
864   2
924   2
936   2
960   2
gp >

We can easily adjust this for more pairs , for example :

gp > for(n=1,5000,v=select(m->(n/m^2<2)*(m^2<n),divisors(n));if(length(v)>=3,print(n,"   ",length(v))))
840   3
1080   3
1260   3
1440   3
1680   3
1980   3
2016   3
2160   3
2340   3
2520   4
2640   3
2880   3
3024   3
3080   3
3120   3
3150   3
3300   3
3360   3
3600   3
3696   3
3780   3
3900   3
3960   3
4032   3
4080   3
4200   3
4320   3
4368   3
4536   3
4560   3
4620   3
4680   3
4800   3
4950   3
gp >

It is interesting to find the smallest solution for $2,3,4,\cdots$ pairs

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  • $\begingroup$ Very good, thanks! $\endgroup$
    – user
    Aug 25 '20 at 9:10
  • $\begingroup$ Thank you, but I forgot a little thing, ruling out the squares. This can easily be achieved by a slight modification. The first number to be ruled out is $3600$. $\endgroup$
    – Peter
    Aug 25 '20 at 9:20

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