Question

The digital root of a number is the number obtained by repeatedly addinga the digits of the number. If the answer is not a one-digit number, add those digits. Continue until a one-digit sum is reached. This one digit is the digital root of the number. For example, the digital root of 98 is 8, since 9 + 8 = 17 and 1 + 7 = 8. Record the digital roots of the first 30 integers and find as many patterns as you can. Can you explain any of the patterns?

Answer 1

9514 1404 393

Answer:

• digital roots are 1, 2, 3, ..., 8, 9, 1, 2, 3, ... (repeating 1-9 pattern)
• the digital root is the mod 9 value of the number except that 9 is used instead of 0
• numbers divisible by 3 have digital roots divisible by 3

Explanation:

The digital roots of the single-digit numbers 1-9 are just those numbers.

For numbers 10-18, the digits are the numbers 1-9, as for the numbers 19-27. In short, the digital root values are sequential, matching the remainder from division by 9 (except that numbers evenly divisible by 9 have a digital root of 9).

We also note that numbers divisible by 3 will have a digital root that is divisible by 3.

So, the patterns we notice are ...

• numbers divisible by 9 have a digital root of 9
• numbers divisible by 3 have a digital root divisible by 3

_____

Additional comment

This is a consequence of our base-10 number system and modulo arithmetic. In a base-N number system, the "digital root" will be the value of the number modulo (N-1), again with the exception that 0 is replaced by (N-1). Divisors of (N-1) will also be divisors of the number if they divide the digital root.