Why is modulo 9 the Digital root of all numbers?

Question

heres what i mean:

132189 –> 1 + 3 + 2 + 1 + 8 + 9 = 24 –> 2 + 4 = 6
132189 % 9 = 6

https://en.wikipedia.org/wiki/Digital_root

In: 7

Answer 1

The nice thing about modulo arithmetic is you can largely delay applying the modulo to whatever numbers you’re working with and the result is valid.

Now, your number, 132189 could be written as the sum of each digit as you did, but you need to multiply by the tens/hundreds/thousands for each number. This is the actual number as you wrote it, without modulo math.

100000 + 3*10000 + 2*1000 + 100 + 8*10 + 9

Now it should be fairly obvious that 9, 99, 999, etc are divisible by 9, or at equivalent to 0 modulo 9. These are also all 1 shy of the numbers 10, 100, 1000 etc. So all those numbers are equivalent to 1 module 9. Since we can apply the modulo arithmetic at any point, I choose to do it to my breakdown above on each individual number. Remember, 10, 100, 1000 etc are all equal to 1.

1 + 3*1 + 2*1 + 1 + 8*1 + 9 = 1 + 3 + 2 + 1 + 8 + 9 = 24 (modulo 9)

I do that all over again to `24` get 6.

In any number system base N, N-1 is the digital root. Humans count with N=10. In hexadecimal (N=16) the digital root would be 15. Indeed, 255 (1 less than 16*16) is `15 * 17` and so on.