Math
Digit Sum Calculator
Type a whole number (up to 200 digits) to get its digit sum and digital root.
The steps list each round of adding digits until a single digit remains, then confirm the answer with the shortcut 1 + (N - 1) mod 9. Because a number and its digit sum leave the same remainder when divided by 9, the page also reads off whether the number is divisible by 3 and by 9.
A whole number up to 200 digits. Commas and a leading minus sign are fine; the sign is ignored when adding digits.
Try an example
Result
Digit sum
—
- Digital root
- —
- Additive persistence
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- Divisible by 3 / 9?
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- Product of digits
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- Number of digits
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the digit sum calculator solves
The digit sum of a number is what you get by adding its digits: for 987654321 that is 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45. Repeating the process on the result (4 + 5 = 9) until one digit is left gives the digital root. The number of rounds needed is the additive persistence. This calculator shows every round and the mod-9 shortcut.
Why digit sums reveal divisibility by 3 and 9
Every power of ten is one more than a multiple of 9 (10 = 9 + 1, 100 = 99 + 1, …), so a number and its digit sum leave the same remainder when divided by 9, and therefore also by 3. If the digit sum is a multiple of 9, so is the number; if it is a multiple of 3, so is the number. That is the classic divisibility rule, and the digital root makes it instant: digital root 9 means divisible by 9; digital root 3, 6, or 9 means divisible by 3.
| Number | Digit sum | Digital root | Divisible by 3? | Divisible by 9? |
|---|---|---|---|---|
| 1,234 | 10 | 1 | no | no |
| 7,155 | 18 | 9 | yes | yes |
| 2,022 | 6 | 6 | yes | no |
| 999,999 | 54 | 9 | yes | yes |
| 1,000,000 | 1 | 1 | no | no |
The digital root shortcut
You do not have to repeat the sums: for any positive integer, the digital root is 1 + (N - 1) mod 9, which is the remainder on division by 9 except that a remainder of 0 becomes 9. The calculator reports both routes so you can see they agree.
How to use it
- Type the whole number. Commas are stripped; a minus sign is ignored for the digits.
- Read the digit sum, the digital root, and how many rounds it took.
- Use the divisibility line to see at once whether the number is a multiple of 3 or 9.
- Open Show the work to see each round written out and the 1 + (N - 1) mod 9 shortcut.
How to read the answer
The digit sum can be large for long numbers; the digital root is always 0 to 9 (0 only for the number 0). A digital root of 9 means the number is divisible by 9; 3, 6, or 9 means divisible by 3; anything else means neither. The additive persistence counts the rounds of summing needed, and the product of digits is 0 whenever any digit is 0.
Common mistakes and edge cases
- Stopping after one round when the digital root was asked for: 45 is the digit sum of 987654321, but the digital root is 9.
- Saying the digital root is the remainder mod 9 without the fix-up: 18 mod 9 = 0, but the digital root of 18 is 9.
- Applying the rule to divisibility by 6 or 12 with digit sums alone; those need an extra check on the last digit(s).
- Adding the digits of a decimal or of a number with its minus sign: digit sums are defined for whole numbers, and the sign is ignored.
- Confusing digit product with digit sum: 12 has sum 3 and product 2.
Worked examples
987654321
Digit sum 45, digital root 9.
Digit sum
45
Divisible by 3 but not 9
2022 → 6.
Digit sum
6
Single digit
7 is already its own digit sum and root.
Digit sum
7
Zero
Digit sum and digital root are both 0.
Digit sum
0
Negative number
The sign is ignored: -4,599 → 27.
Digit sum
27
Large number
Thirty 9s sum to 270, root 9, persistence 2.
Digit sum
270
Decimal input
Not allowed.
Digit sum
Error
Frequently asked questions
What is the digit sum of a number?+
The total you get by adding all its digits. The digit sum of 4,599 is 4 + 5 + 9 + 9 = 27.
What is a digital root?+
The single digit you reach by repeatedly summing digits: 4,599 → 27 → 9. It equals 1 + (N - 1) mod 9 for positive N.
How does the digit sum tell me if a number is divisible by 9?+
A number and its digit sum have the same remainder when divided by 9. So if the digit sum (or digital root) is a multiple of 9, the number is too. 7,155 has digit sum 18, so it is divisible by 9 (7,155 = 9 × 795).
Does the digit sum test work for divisibility by 3?+
Yes, for the same reason: if the digit sum is divisible by 3, so is the number. 2,022 has digit sum 6, so 2,022 is a multiple of 3 (674 × 3).
What is additive persistence?+
The number of digit-sum rounds needed to reach a single digit. 987654321 → 45 → 9 takes two rounds, so its additive persistence is 2.
Can the digital root be 0?+
Only for the number 0 itself. Every positive integer has digital root 1 through 9.
What is casting out nines?+
An error check for arithmetic: replace each number by its digital root, redo the operation, and compare digital roots. If they differ, the original calculation is wrong.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026