Enter any positive integer into the Digital Root Calculator and find its digital root — the single digit you get by repeatedly summing all digits until only one remains. You'll see the digital root value, the number of reduction steps, and a full step-by-step breakdown of each iteration.
Digital Root
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Number of Steps
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First Digit Sum
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A digital root is the single-digit value obtained by repeatedly summing all the digits of a number. You keep adding the digits of each resulting number until you reach a single digit between 1 and 9 (with 0 being the digital root only for the number 0).
The calculator takes your input number and sums its digits. If the result is more than one digit, it repeats the process on the new number. This continues until a single digit remains. For example, 56984 → 5+6+9+8+4 = 32 → 3+2 = 5, so the digital root is 5.
Yes. For any positive integer n, the digital root equals n − 9 × (ceil(n / 9) − 1), which simplifies to: if n is divisible by 9, the digital root is 9; otherwise it is n mod 9. This gives the same result as repeated digit summation.
Digital roots are used to verify arithmetic calculations — a technique known as 'casting out nines.' They also appear in number theory, recreational mathematics, and certain divisibility tests. If the digital root of a number is 3, 6, or 9, the number is divisible by 3.
The digital root of 0 is 0. It is the only case where the digital root is not between 1 and 9.
The ceiling function (ceil) rounds a number up to the nearest integer. In the digital root formula, ceil(n / 9) is used to determine how many multiples of 9 fit into or just above the number, helping derive the digital root without iteration.
Yes. Every positive integer has a digital root in the range 1 to 9. The process of repeated digit summation always converges to a single digit, and since single digits already are their own digital roots, the result is always 1–9 (or 0 for the input 0).
A number is divisible by 9 if and only if its digital root is 9. Similarly, a number is divisible by 3 if its digital root is 3, 6, or 9. This makes digital roots a quick mental check for divisibility.