-2
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List 1 | List 2 | List 3

 10        7        5
 20       11       22
 34       32       29
 72       59       33
 80       81       83
 96       99      107
130      140      150
132      190      160

The numbers are picked randomly,
There are more possible numbers for List 2 than for List 1.
List 3 contains random numbers with no common properties.

Can you figure out the property the numbers in List 1 share?
Can you place the numbers from List 3 into Lists 1 & 2 according to their properties?

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  • $\begingroup$ When you say "There are more possible numbers for List 2 than for List 1" do you mean that List 1 is finite and smaller than List 2, or merely that List 1 has fewer members (than List 2) that are less than any given, large number? $\endgroup$ – msh210 Oct 10 at 10:25
  • $\begingroup$ @msh210 Good question. The second: List 1 has fewer members (than List 2) that are less than any given, larger number. $\endgroup$ – Nati Oct 10 at 11:46
  • $\begingroup$ I don't understand the question: "Can you sort the numbers of List 3 into the other two list?". What do you want us to do here? $\endgroup$ – Dmitry Kamenetsky Oct 11 at 1:31
  • $\begingroup$ @DmitryKamenetsky Figure out the property of List 1&2 and put the numbers of List 3 in List 1 and List 2 according to the property. $\endgroup$ – Nati Oct 11 at 6:59
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Well, there must be an almost-infinite number of possible solutions, so I'll post my one (which is probably far from intended answer, but still not too far-fetched to match this exact question, I believe).

- List 1 contains all even numbers, which either have an even number of prime divisors, or have at least one prime divisor containing digit 3 (in decimal expansion): 10, 20, 34, 72, 80 and 96 all have 2 prime divisors each (2 and 5, 2 and 5, 2 and 17, 2 and 3, 2 and 5, 2 and 3 respectively), while 130 and 132, despite having 3 prime divisors (2,5,13 and 2,3,11 respectively), can be divided by a prime which has a 3 in it (3 and 13, respectively).
- On the other hand, List 2 contains all other numbers (i.e. all odd numbers and even numbers with an odd count of prime divisors, neither of them containing a 3), e.g. 7, 11, 59, 81, 99 are all odd, while 32 ($=2^5$), 140 ($=2^2\times5\times7$) and 190 ($=2\times5\times19$) contain an odd number of prime divisors, neither of them has a 3 in its decimal expansion.

Note that

there are more possible numbers for List 2 than for List 1 (from a set of positive integers not greater than some given number $N$, i.e. $\{1,2,\dots,N\}$), because the latter contains only part of the even numbers, while the former contains all odd and some even ones (and the count of even numbers in $\{1,2,\dots,N\}$ is never greater than the count of odd ones).

So

22 ($=2\times11$ - 2 prime divisors), 150 ($=2\times3\times5^2$ - 3 prime divisors but including the 3) and 160 ($=2^5\times5$ - 2 prime divisors) go into List 1, while all other numbers - 5, 29, 33, 83 and 107 (being all odd) - go into List 2.

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