Divisibility Test Question of Curisosity

Question

Why do we only do divisibility tests up to 11? At least, in my proofs class and in my textbook, that's all it goes up to: 11.

Can anyone explain?

Answer 1

First, in most cases the divisibility test for numbers $\le 11$ are quite easy.

On the other hand, the divisibility test for larger numbers are either a combination of previous ones (a number is divisible by $15$ iff it's divisible by both $3$ and $5$) or are quite tiresome to check. There're exceptions, of course.

Finally, you don't really need those tests. By the time when you might need to apply them, your skill are already good enough to check the divisibility by hand (or in your mind), or you already memorise the multiples of this number.

Answer 2

Every number from $2$ up to $13^2-1 = 168$, is either a prime number or divisible by $2,3,5,7,$ or $11$. Realistically, is it worth to effort to test $19238820482$ for divisibility by $7$ or, say, $47$ by hand?