Adding the digits of a number seems like a trivial operation, but it carries real information because of how base ten works. Ten leaves a remainder of one when divided by nine, and so does every power of ten. That means a number and its digit sum always leave the same remainder when divided by nine, which is why the digit sum tests divisibility by nine and by three.
Repeat the digit sum until one digit remains and you have the digital root. It equals the number modulo nine, with nine standing in for zero. Accountants once used this as casting out nines, a quick check on long addition and multiplication: if the digital roots of the operands do not combine to the digital root of the answer, there is an error somewhere. It catches most slips, though not a transposition, since swapping digits leaves the sum unchanged.
The formula
n- The number whose digits are being added
Σ digits- The digits added once
digital root- The result of repeating that until one digit remains
mod 9- The remainder on division by nine, which the digital root reproduces
How it works, step by step
- Enter a whole number.
- The gauge shows the sum of its digits.
- The digital root below reduces that to a single digit.
- A digit sum divisible by three or nine means the number is too.
Worked examples
987,654 digit by digit
9 + 8 + 7 + 6 + 5 + 4 = 39, and 39 reduces to 12 then to 3. A digital root of 3 means the number is divisible by three but not by nine, and indeed 987654 ÷ 3 = 329,218.
Casting out nines as a check
To check 4,821 × 37 = 178,377, take digital roots: 4,821 gives 6 and 37 gives 1. Their product 6 has digital root 6, matching the root of 178,377, which is 6. The answer survives the check.
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Frequently asked questions
Why does the digit sum test divisibility by nine?
Because every power of ten leaves remainder one when divided by nine. A number is therefore congruent to the sum of its digits modulo nine, so if the digit sum divides by nine, so does the number.
What is a digital root used for?
Casting out nines — a fast sanity check on hand arithmetic. It also appears in checksum schemes, though modern ones use weighted sums because a plain digit sum misses transpositions.
Does it catch every arithmetic error?
No. It misses any error that changes the value by a multiple of nine, including swapping two digits. It is a filter that catches most slips cheaply, not a proof of correctness.
Is there a shortcut to the digital root?
Yes: it is 1 + ((n − 1) mod 9) for any positive n, so no repeated summing is needed. Zero is the only number with a digital root of zero.
Digit sums and roots
| Number | Digit sum | Digital root | Divisible by |
|---|---|---|---|
| 12 | 3 | 3 | 3 only |
| 99 | 18 | 9 | 9 and 3 |
| 123 | 6 | 6 | 3 only |
| 999 | 27 | 9 | 9 and 3 |
| 4,821 | 15 | 6 | 3 only |
| 987,654 | 39 | 3 | 3 only |
| 1,000,000 | 1 | 1 | neither |
The digital root equals the number modulo nine, with nine used in place of zero.