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 number 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 two 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 Congruence relations 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:

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.

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