Digital Root - Congruence Formula

Congruence Formula

The formula is:

or,

To generalize the concept of digital roots to other bases b, one can simply change the 9 in the formula to b - 1.

The digital root is the value modulo 9 because and thus so regardless of position, the value mod 9 is the same – – which is why digits can be meaningfully added. Concretely, for a three-digit number,

To obtain the modular value with respect to other numbers n, one can take weighted sums, where the weight on the kth digit corresponds to the value of modulo n, or analogously for for different bases. This is simplest for 2, 5, and 10, where higher digits vanish (since 2 and 5 divide 10), which corresponds to the familiar fact that the divisibility of a decimal number with respect to 2, 5, and 10 can be checked by the last digit (even numbers end in 0, 2, 4, 6, or 8).

Also of note is since and thus taking the alternating sum of digits yields the value modulo 11.