Considering the whole numbers from 1 to 12, I divided them into four sets:
$A=\{1,5,9\};\\ B=\{2,6,10\};\\ C=\{3,7,11\};\\ D=\{4,8,12\}.$
It is observed that $A$ and $C$ are odd numbers, while $B$ and $D$ are even numbers. $B$ and $D$ differ in that $B$'s numbers halve into odd numbers ($2/2=1$; $6/2=3$; $10/2=5$), whereas $D$'s numbers in even numbers ($4/2=2$; $8/2=4$; $12/2=6$). Similarly, I am wondering whether there is a feature of the set $A$ that differentiates it from the set $C$. I have found a possible solution through Modular Arithmetic, but I think there may be a simpler solution that I cannot find, which is limited to the realm of Elementary Mathematics. To be more precise, my question is as follows: is there an arithmetic property that differentiates the numbers of $A$ from those of $C$?
Keep going a bit - what happens to $13,14,15,16$ and so on. Does each of your sets above keep getting bigger or do you need more sets $E,F$ and so on to describe this process? Does the process go on indefinitely? You've already mentioned modular arithmetic and the previous comments pick up on that.
One way of describing this situation a little more formally is that each of your resulting sets are equivalence classes (two numbers are equivalent if they have the same remainder upon dividing by $4$) and taking it a little further, these are the elements in the cyclic group $\mathbb{Z}/4\mathbb{Z}$ and are called cosets in that context.
If this is not familiar to you yet that is fine, but you'll get a sense of where this is heading in a course or two (or three).