Digital Root Calculator
Sum the digits until one is left. It is always n mod 9.
Work out Digital Root. Sum the digits until one is left. It is always n mod 9. States the assumption instead of hiding it.
Digital root
2
9875 → 29 → 11 → 2 · 3 steps
Sum the digits, then sum the digits of that, until one digit remains. The result is always the number modulo 9, with 9 standing in where the remainder would be 0 — which is not a coincidence but the whole reason the trick works. That identity is why the schoolroom divisibility tests hold: a number is divisible by 3 exactly when its digital root is 3, 6 or 9, and by 9 exactly when its digital root is 9. Both tests are the mod-9 fact stated differently. Casting out nines, the old method of checking arithmetic by hand, is the same idea again. It catches most errors cheaply but misses any mistake that happens to be a multiple of 9 — including digit transpositions, which is a real limitation rather than a footnote.
How the Digital Root Calculator works
The digital root of any whole number, with every reduction step shown and the divisibility tests that follow from it. The result is always n modulo 9 with 9 standing in for zero — which is precisely why the digit-sum tests work.
Also known as: sum of digits until one digit · is it divisible by 9 · casting out nines · repeated digit sum calculator
Frequently asked questions
What is a digital root?
Sum a number's digits, then sum the digits of the result, and repeat until one digit remains. For 9875 that is 29, then 11, then 2 — three steps to reach a digital root of 2.
Why does the digit-sum test for divisibility by 3 work?
Because the digital root equals the number modulo 9. Any number divisible by 3 has a digital root of 3, 6 or 9, and any number divisible by 9 has a digital root of exactly 9. The tests are that identity restated.
What is casting out nines?
An old method of checking arithmetic by hand: compute the digital roots of the inputs, apply the same operation, and compare with the digital root of the answer. It catches most errors quickly and cheaply.
Are there errors casting out nines misses?
Yes — any mistake that happens to differ from the correct answer by a multiple of 9. Transposed digits are the important case, since swapping two digits always changes the value by a multiple of 9 and so passes the check undetected.
What is additive persistence?
The number of summing steps needed to reach a single digit — three for 9875. It grows extremely slowly: the smallest number requiring four steps is 19,999,999,999,999,999,999,999.
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