# Which number belongs where?

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?

• 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? – msh210 Oct 10 at 10:25
• @msh210 Good question. The second: List 1 has fewer members (than List 2) that are less than any given, larger number. – Nati Oct 10 at 11:46
• 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? – Dmitry Kamenetsky Oct 11 at 1:31
• @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. – Nati Oct 11 at 6:59

- 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.
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).
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.