Digital Root Calculator

10≡1 (mod 9), so 10ᵏ≡1 (mod 9). Any number n = Σdᵢ·10ⁱ ≡ Σdᵢ (mod 9). The digit sum preserves the value mod 9. Since dr(n)∈{1,...,9} and dr(n)≡n (mod 9), we get dr(n) = 1+((n−1) mod 9) for n>0.

The number of digit-sum steps to reach a single digit. Most numbers have low persistence. The smallest number with persistence 1 is 10, persistence 2 is 19, persistence 3 is 199, persistence 4 is 19999999999999999999999. No number with persistence >11 is known.

To check a×b=c: compute dr(a)×dr(b) and compare with dr(c). If they differ (mod 9), there's an error. Example: 123×456=56088. dr(123)=6, dr(456)=6, 6×6=36→dr=9. dr(56088)=27→9. Matches! Fast mental arithmetic check.