Divisibility is not about remainders — it is about structure. When b divides a, it means a is built from b. Everything in number theory starts here.
Divisibility rules are not tricks to memorise — they are theorems about what happens to a number's digits modulo small primes. Each rule has a one-line proof.
How many divisors does n have? What is their sum? The answers come from the prime factorization of n, and the formulas are cleaner than you'd expect.