Let’s try 13:
1 + 3 = 4
Four is the digit sum of 13. It should be the remainder you would get if you divided by 9. Nine
divides into 13 once, with 4 remainder.
If you add 3 to the number, you add 3 to the remainder. If you double the number, you double the
remainder. If you halve the number you halve the remainder.
Don’t believe me? Half of 13 is 6.5. Six plus 5 equals 11. One plus 1 equals 2. Two is half of 4, the
nines remainder for 13.
Whatever you do to the number, you do to the remainder, so we can use the remainders as
substitutes.
Why do we use 9 remainders? Couldn’t we use the remainders after dividing by, say, 17? Certainly,
but there is so much work involved in dividing by 17, the check would be harder than the original
problem. We choose 9 because of the easy shortcut method for finding the remainder.