The Magical 9-Test!

Ever wondered if a huge number like 78,923,415 is perfectly divisible by 9 without actually dividing it? There's a secret trick!

Subject: number-theory • Classes: 6–12 • Difficulty: intermediate (9-10)

The Trick

To quickly check if any number is divisible by 9, simply sum all its digits. If this sum is divisible by 9, then the original number is also divisible by 9! This works because $10^k \equiv 1 \pmod{9}$. So, any number $N = a_n 10^n + ... + a_1 10^1 + a_0 10^0$ can be written as $N \equiv (a_n + ... + a_1 + a_0) \pmod{9}$. Thus, $N$ and its sum of digits have the same remainder when divided by 9.

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