๐Ÿ”„ Digital Root Calculator

Calculate the digital root of a positive integer using repeated modulo-9 arithmetic.

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Fill in the details and click Calculate to see your results.


Also available in our Android app, with charts and full breakdown tables.

๐Ÿ“˜ About the Digital Root

The digital root repeatedly adds the digits until one digit remains.

For positive integers, it can be obtained directly from the number modulo 9.

๐Ÿงฎ Formula

Modulo 9
r = n mod 9
Digital root rule
root = 9 when r = 0

Where

  • n = positive integer
  • r = n mod 9
  • digital root = 9 if r = 0, otherwise r
This shortcut applies to positive integers. Zero has digital root 0, but this calculator is designed for positive inputs.

โœ๏ธ Worked example

Digital root of 9875

You enter
Positive Integer9875
You get
Digital Root2
Remainder Modulo 92

9875 modulo 9 is 2, so the digital root is 2.

Try this example โ†’

๐Ÿงญ How to use this calculator

What you enter

  • Positive Integer

What you get

  • Digital Root
  • Remainder Modulo 9

๐Ÿ’ก Uses

  • Number puzzles and arithmetic checks.
  • A quick divisibility-by-9 related calculation.
  • Studying properties of decimal digits.

โ“ Frequently asked questions

What is a digital root?

It is the single-digit value obtained by repeatedly summing the digits.

Why does modulo 9 work?

In base 10, powers of 10 leave remainder 1 modulo 9.

What is the digital root of 9?

It is 9 for the positive-integer convention.