The digital root (also repeated digital sum) of a number is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum. The process continues until a single-digit number is reached.
For example, the digital root of is , because and
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 the summands' digital roots. A person 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.
Digital roots are used in Western numerology, but certain numbers deemed to have occult significance (such as 11 and 22) are not always completely reduced to a single digit.
The number of times the digits must be summed to reach the digital sum is called a number's additive persistence; in the above example, the additive persistence of 65,536 is 2.
Significance and formula of the digital root 
It helps to see the digital root of a positive integer as the position it holds with respect to the last multiple of nine less than it. For example, the digital root of 11 is 2, which means that 11 is the second number after 9. The digital root of 2035 is 1, which means that 2035-1 is a multiple of nine. A digital root of nine means that the number is a multiple of nine, which is equivalent to a digital root of 9.
With this in mind the digital root of a positive integer may be defined by using Floor function , as
Abstract multiplication of digital roots 
The table below shows the digital roots produced by the familiar multiplication table in the decimal system.
Formal definition 
Let denote the sum of the digits of and let the composition of as follows:
Eventually the sequence becomes an one digit number. Let (the digital sum of ) represent this an one digit number.
Let us find the digital sum of .
For simplicity let us agree simply that
Proof that a constant value exists 
How do we know that the sequence eventually becomes an one digit number? Here's a proof:
Let , for all , is an integer greater than or equal to 0 and less than 10. Then, . This means that , unless , in which case is an one digit number. Thus, repeatedly using the function would cause to decrease by at least 1, until it becomes an one digit number, at which point it will stay constant, as .
Congruence formula 
The formula is:
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.
Some properties of digital roots 
The digital root of zero is zero if and only if the number is itself zero.
The digital root of a positive integer is a positive integer if and only if the number is itself a positive integer.
The digital root of n is n itself if and only if the number has exactly one digit.
The digital root of n is less than n if and only if the number is greater than or equal to 10.
The digital root of a + b is congruent with the sum of the digital root of a and the digital root of b modulo 9.
The digital root of a - b is congruent with the sub of the digital root of a and the digital root of b modulo 9.
The digital root of a × b is congruent with the multiple of the digital root of a and the digital root of b modulo 9.
- The digital root of a nonzero number is 9 if and only if the number is itself a multiple of 9.
- The digital root of a nonzero number is a multiple of 3 if and only if the number is itself a multiple of 3.
- The digital root of a factorial ≥ 6! is 9.
- The digital root of a square is 1, 4, 7, or 9. Digital roots of square numbers progress in the sequence 1, 4, 9, 7, 7, 9, 4, 1, 9.
- The digital root of a perfect cube is 1, 8 or 9, and digital roots of perfect cubes progress in that exact sequence.
- The digital root of a prime number (except 3) is 1, 2, 4, 5, 7, or 8.
- The digital root of a power of 2 is 1, 2, 4, 5, 7, or 8. Digital roots of the powers of 2 progress in the sequence 1, 2, 4, 8, 7, 5. This even applies to negative powers of 2; for example, 2 to the power of 0 is 1; 2 to the power of -1 (minus one) is .5, with a digital root of 5; 2 to the power of -2 is .25, with a digital root of 7; and so on, ad infinitum in both directions. This is because negative powers of 2 share the same digits (after removing leading zeroes) as corresponding positive powers of 5, whose digital roots progress in the sequence 1, 5, 7, 8, 4, 2.
- The digital root of an even perfect number (except 6) is 1.
- The digital root of a star number is 1 or 4. Digital roots of star numbers progress in the sequence 1, 4, 1.
- The digital root of a triangular number is 1, 3, 6 or 9. Digital roots of triangular numbers progress in the sequence 1, 3, 6, 1, 6, 3, 1, 9, 9.
- The digital root of Fibonacci numbers is a repeating pattern of 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9.
- The digital root of Lucas numbers is a repeating pattern of 2, 1, 3, 4, 7, 2, 9, 2, 2, 4, 6, 1, 7, 8, 6, 5, 2, 7, 9, 7, 7, 5, 3, 8.
- The digital root of the product of twin primes, other than 3 and 5, is 8. The digital root of the product of 3 and 5 (twin primes) is 6.
See also 
- F. M. Hall: An Introduction into Abstract Algebra. 2nd edition, CUP ARchive 1980, ISBN 978-0-521-29861-2, p. 101 (online copy at Google Books)
- Bonnie Averbach, Orin Chein: Problem Solving Through Recreational Mathematics. Courier Dover Publications 2000, ISBN 0-486-40917-1, pp. 125-127 (online copy at Google Books)
- Talal Ghannam: The Mystery of Numbers: Revealed Through Their Digital Root. CreateSpace Publications 2012, ISBN 978-1477678411, pp. 68-73
- T. H. O'Beirne: Puzzles and Paradoxes. In: New Scientist, No. 230, 1961-4-13, pp. 53-54 (online copy at Google Books)¨