How do mathematicians figure stuff like divisibility. Like, if

>>8963544
11*

First it starts with an observation
>Hey look the first few multiples of 1 have the sum of this property

Then you think of some way to prove it.

In your case it's not even hard, it's just a simple proof by induction.
If a number has the same sum of the digits in the odd and even place, then adding 11 to it also does. Since that's true for 0, it has to be true for any 11*k with k natural.

Viceversa, the same property is preserved by subtracting 11, so for any number k' that has the property you can subtract it till you reach a number lesser than 11, which has to be 0 because it is the only one with that property (otherwise adding 11 back the same amount of time can't bring you back to the starting number for the first property). Therefore k' is a multiple of 11.

I do agree that harder examples seem kinda random but when you study the matter more you start to see more order. Of course we can't do everything as human beings, some theorems are indeed proved by brute force.

>>8963544

Bullshit. 11*9173 = 100903. The alternate digits don't add up.

Checkmate atheists.

>>8963552
>>8963568
whoops, seems like I fucked up
I didn't think of the cases with 9's in it

the general thought process is still the same, you make conjectures and then try to see if they're true, and if you become fairly sure you try to make a proof

>>8963544
It's really a lot of fiddling with the base-10 number system, which follows from the fact that any number in base-10 is just the addition of the digits multiplied by their respective power of 10. Ones place is multiplied by 10^0, tens place is multiplied by 10^1, and so on. From that, you can probably figure out proofs yourself with just pencil and paper.

Seeing if a number is divisible by two is easy, just look at the last digit and know the multiplication table of two for single digit numbers.
I assume there are similar "tricks" for divisibility with other numbers as well, they just might have more steps, bigger but still manageable multiplication tables.

>>8963568
> 11*9173 = 100903. The alternate digits don't add up.

That can be fixed by making a modification to OP's claim:

A number is divisible by 11 if the following two numbers are equal:
* the sum of its even-place digits taken mod 11
* the sum of its odd-place digits taken mod 11

For example, 418 is divisible by 11 because (12 mod 11) = (1 mod 11)