Digital sum/root

Digital sum – the sum of all its digits. (12345 => 1+2+3+4+5=>15).

Digital root (also repeated digital sum) – the sum of all its digits untill a single-digit number is reached.
12345 => 1+2+3+4+5 => 15 => 1+5 => 6

In other words, it is the remainder when the number is divided by 9.

12345/9 => 12339 with a remainder of 6 => 1371.66 (1371*9=>12339 +6 => 12345).

Casting Out 9’s (a fast to find digital root).
Ignore the digit 9 in the number or a 9 anywhere in the number.

129 => 12 (remove 9) => 3 (1+2+9 => 12 => 3).

12999999993 => 1+2+3 => 6 (1+2+9+9+9+9+9+9+9+9+3 => 78 => 15 => 6).

1234125678 => (cross it out 1+8,2+7,3+6,4+5), so remaining 1,2 => 1+2 => 3
=> 1+2+3+4+1+2+5+6+7+8 => 39 => 12=> 3.

What is the purpose of the Digital Sum?
when the numbers are added, subtracted, multiplied or divided; their digital sums will also be respectively added, subtracted, multiplied or divided. So, digital sums to check the accuracy of the answer (mostly for big numbers).

123456789 * 123456789 = 15241578750190521
0* 0 = 115515 => 1+1+5+5+1+5 => 18 => 9 => 0 (so it correct)

123456789 * 123456789 = 15241578750190522 (change last digit 1 to 2).
0* 0 = 115552 => 1+1+5+5+5+2 => 19 => 10 => 1 (so it is not correct).

But it won’t find all the mistakes, let us the above exmaple, added 0 end of the result

123456789 * 123456789 = 152415787501905210(added 0, answer is wrong, but digit sum is correct)
0* 0 = 115515 => 1+1+5+5+1+5 => 18 => 9 => 0 (so it correct).

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