Repeatedly sum the digits of a nonnegative integer until only one digit remains. Each submitted value produces digital root plus the intermediate reasoning.
The digital root compresses a base-ten integer to one digit while preserving its remainder modulo 9, with positive multiples of 9 represented by 9 rather than 0.
Preserve Nonnegative integer when checking the Digital Root output. Keep the fixed condition under the same convention and estimate Digital root. A controlled change to the fixed condition should move the Digital Root Digital root in a mathematically consistent direction.
The digits of 987654321987654321 sum to 90; 9 + 0 gives digital root 9.
Digit order does not matter to the result, so transposed digits often escape this check. Zero is assigned digital root zero.
An impossible digital root sign or magnitude should prompt a Digital Root input review before rounding.
Add every digit. If the total has more than one digit, repeat until a single digit remains.
Digit checks, recreational number theory, and casting-out-nines verification use this invariant. It can detect many arithmetic errors but cannot prove a calculation correct. A related application of Digital Root is casting out nines.
Digital roots also reveal divisibility by three: roots 3, 6, or 9 indicate a multiple of three. This works because ten and every power of ten leave remainder one when divided by three, so replacing place values with their digits preserves that remainder.
Substitute the Digital Root solution into Nonnegative integer. This Digital Root check rejects false Digital Root branches and forbidden denominators.
Choose an easy Nonnegative integer value before running Digital Root. Predict the result, then compare it with the Digital Root output.
An approximate Digital root offers a separate test of Digital Root. Derive that estimate from Nonnegative integer while retaining the fixed condition. Exact agreement is unnecessary; the estimate only needs enough accuracy to expose an implausible Digital Root magnitude.
Yes, many numbers share each residue class modulo 9.
Yes; a positive integer is divisible by 9 when its digital root is 9.
Each decimal place is congruent to 1 modulo 9.
Zero.