Digital Roots & Casting Out Nines

Before calculators, merchants checked huge sums by 'casting out nines' — add a number's digits, reduce mod 9, and compare. If your ledger's two sides disagree mod 9, you've made an error. It's not magic: it's the fact that a number and the sum of its digits are always congruent mod 9. Repeat the summing down to a single digit and you get the digital root — mod 9 wearing a friendlier face.