Digital root

The digital root or digital sum of a non-negative integer 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 the digit sum. The process continues until a single-digit number is reached.

The digital root of a number is equal to the remainder when that number is divided by 9. If the remainder is 0 and the number is greater than 0, then the digital root is 9. If the number is 0, then the digital root of the number is 0.

For example, the digital root of the number 12,345 is 6 because 1 + 2 + 3 + 4 + 5 equals 15 and then 1 + 5 equals 6.

A shortcut to finding digital root:

Casting out 9’s:

Consider the number 27518. Its digital root is 5 because 2 + 7 + 5 + 1 + 8 = 23 and 2 + 3 = 5.

Now, the digital sum of 27518 can also be calculated by simply grouping (2 + 7) + 5 + (1 + 8), where the un-grouped value 5 is the digital root (since 2 + 7 = 9 and 1 + 8 = 9 are eliminated from the calculation).

Properties of digital root:

• If we multiply any number by 9, the digital root will always be 9.
• Adding 9 to a number does not change the digital root of that number.
• If we divide any number by 9, the digital root of that number will be the remainder.

Multiplication Magic with 9:

We know that

$$9 \equiv 0 \pmod{9}.$$

Clearly,

$$9x \equiv 0 \pmod{9}.$$

When a positive integer $x$ is multiplied by 9, the digital root will be always equal to 9 because 9 is the factor of the product. Clearly, the sum of digits of a product of 9 is divisible by 9. A number is divisible by 9 if and only if the sum of digits of the number is divisible by 9.

Here is an example:

$$(1234)(9) \equiv 0 \pmod{9}.$$

Addition Magic with 9:

We know that

$$N \equiv x \pmod{9},$$

where $N$ is any positive integer and $x$ is the digital root of $N.$ Since $9 \equiv 0 \pmod{9},$

$$\begin{aligned} N+9 &\equiv 0+x \pmod{9}\\ N+9 &\equiv x \pmod{9}. \end{aligned}$$

Hence, adding 9 to a number will not affect its digital root.

Here is an example:

$$\begin{aligned} 1234 &\equiv 1 \pmod{9}\\ 9 &\equiv 0 \pmod{9}\\ 1234+9 &\equiv 1 \pmod{9}. \end{aligned}$$

Application of digital root:

When the numbers are added, subtracted, multiplied, or divided, their digital sums will also be added, subtracted, multiplied, or divided, respectively. So, the digital sums can be used to check the accuracy of the answer.

Validating Addition:

$$45723+36245=81,968$$

Clearly, the digital root of 45723 is 3, the digital root of 36245 is 2, and the digital root of 81,968 is 5.
Now, 45723 + 36245 = 81,968, which is validated using the digital root as 3 + 2 = 5.

Validating Subtraction:

$$45723-36245=9,478$$

Clearly, the digital root of 45723 is 3, the digital root of 36245 is 2, and the digital root of 9,478 is 1.
Now, 45723 - 36245 = 9,478, which is validated using the digital root as 3 - 2 = 1.

Validating Multiplication:

$$185 \times 762= 140,970$$

Clearly, the digital root of 185 is 5, the digital root of 762 is 6, and the digital root of 140,970 is 3.
Now, 185 x 762 = 140,970, which is validated using the digital root as 5 x 6 = 30 and the fact that digital root of 30 is 3.

Validating Division:

$$\frac{18,732 }{223}= 84\quad \text{ which is equal to }\quad 223 \times 84= 18,732$$

Clearly, the digital root of 223 is 7, the digital root of 84 is 3, and the digital root of 18,732 is 3.
Now, 223 x 84 = 18,732, which is validated using the digital root as 7 x 3 = 21 and the fact that digital root of 21 is 3.