Fast divisibility tests (by 2,3,4,5,.., 16)?

Question

What are the fastest divisibility tests? Say, given a little-endian architecture and a 32-bit signed integer: how to calculate very fast that a number is divisible by 2,3,4,

Answer 1

In a previous question, I showed a fast algorithm to check in base N for divisors that are factors of N-1. Base transformations between different powers of 2 are trivial; that's just bit grouping.

Therefore, checking for 3 is easy in base 4; checking for 5 is easy in base 16, and checking for 7 (and 9) is easy in base 64.

Non-prime divisors are trivial, so only 11 and 13 are hard cases. For 11, you could use base 1024, but at that point it's not really efficient for small integers.