The divisible-by-9 magic trick

Question

A friend claims they can tell whether any number is divisible by 9 in seconds, just by glancing at it. Take the number 918273645.

Is it divisible by 9? And why does adding up a number's digits reveal divisibility by 9?

Accepted Answer

Yes, 918273645 is divisible by 9. The trick. Add the digits: 9 + 1 + 8 + 2 + 7 + 3 + 6 + 4 + 5 = 45. Since 45 is divisible by 9 (it's 9 	imes 5), the original number is too. (If in doubt, add again: 4 + 5 = 9, clearly divisible.) A number is divisible by 9 exactly when its digit sum is divisible by 9. Why it works. Each place value, 10, 100, 1000, and so on, is just 1 more than a multiple of 9: 10 = 9 + 1,\quad 100 = 99 + 1,\quad 1000 = 999 + 1,\ \dots So a digit d in, say, the hundreds place contributes d 	imes 100 = d 	imes 99 + d. The "99" part is already a multiple of 9 and contributes…

The divisible-by-9 magic trick

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