The digital root, can it be useful for anything?
I've read that the digital root can be used to check the correctness of multiplication. But my feeling is that its very error prone and can not always be exact.
I wonder why the digital root is even defined, must have some sort of usage, but is it just there for games and fun or can it be useful?
Here is an example of the application on multiplication: $$12345\cdot3451=42672595$$
The digital root of lefthand side:
multiplicand $12345$ is $1+2+3+4+5=15\rightarrow1+5=6$.
multiplier $3451$ is $3+4+5+1=13\rightarrow1+3=4$.
The digital root of lefthand side is $6\cdot4 = 24$ is $1+4 = 5$.
The digital root of righthand side:
$4+2+6+7+2+5+9+5 = 40 = 4+0 = 4$.
Compare left and right side: $$5 \neq 4$$
hence the multiplication is incorrect.
To reiterate the basic question I have is, will the digital root always be useful or will it be too error prone to be useful?
The digital root corresponds to reducing the calculation $\bmod 9$. The integers $\bmod 9$ form a ring, so addition, subtraction, and multiplication work as you would expect. If the original calculation is correct the digital root calculation will also be correct, but it doesn't work in the other direction. As Matthew Daly comments, swapping two digits will not change the digital root, so you will not catch errors which swap digits. Another example is carrying into the wrong place. If you add $123$ and $689$, when you add $3+9=12$ you should carry the $1$ into the tens place. If you add it into the hundreds instead, you get $902$ instead of $812$, both of which have a digital root of $2$ and you would not detect the error.