Digital root

The digital root of a number is the number received by adding all the digits, then adding the digits of that number, and then continuing until a single-digit number is reached.

For example, the digital root of 65,536 is 7, because $6+5+5+3+6=25$ and $2+5=7$

Special cases of digital roots of particular numbers are:

Digital root of a square is 1, 4, 7, or 9
Digital root of a perfect cube is 1, 8 or 9
Digital root of a prime number (except 3) is 1, 2, 4, 5, 7, or 8
Digital root of a power of 2 is 1, 2, 4, 5, 7, or 8
Digital root of a perfect number (except 6) is 1
Digital root of a star number is 1 or 4
Digital root of a triangular number is 1, 3, 6 or 9
Digital root of a factorial ≥ 6! is 9.

Digital roots can be calculated with congruences rather than by adding up all the digits, a procedure that can be a real time saver in the case of very large numbers.

The formula is:

{\mbox{dr}}(n)={\begin{cases}n\ ({\rm {mod}}\ 9)\ n\ \neq 0\ ({\rm {mod}}\ 9)\\9\ \ \ \ \ \ \ \ \ \ \ \ \ n\ \equiv 0\ ({\rm {mod}}\ 9)\end{cases}}

{\mbox{dr}}(n)=1\ +\ [n-1({\rm {mod}}\ 9)]

Digital roots can be used as a sort of checksum. For example, since the digital root of a sum is always equal to the digital root of the sum of each summand's digital root, somebody adding long columns of large numbers will often find it reassuring to apply casting out nines to his or her result — knowing that this technique will catch the majority of errors.

See also