The Digit Sum Divisibility by 9

Ever wonder if a huge number like $78,453$ is perfectly divisible by 9 without actually dividing? There's a super-speedy secret that uses just addition!

Subject: number-theory • Classes: 6–12 • Difficulty: basic

The Trick

The 'Digit Sum Trick' for 9: A number is divisible by 9 if the sum of its individual digits is divisible by 9. If the sum is still large, just sum its digits again! \n\nWHY it works: Any number, say $abc$, can be written as $100a + 10b + c$. This can be rewritten as $(99a + a) + (9b + b) + c = 99a + 9b + (a+b+c)$. Since $99a$ and $9b$ are always divisible by 9, the divisibility of the original number $abc$ depends only on whether $(a+b+c)$ (the sum of its digits) is divisible by 9.

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