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.

An example is:

The digital root of 65,536 is 6+5+5+3+6 = 25, then 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 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.

Digital roots can be used as a sort of checksum.

See also:

digit sum
cube (arithmetic)