Casting Out Nines (Checking)

Mental Math · Axiom Academy

A two-second check that catches most arithmetic slips without redoing the sum. 1. Shrink Any Number to One Digit Take any number and add up its digits . If the total still has more than one digit, add those digits too, and keep going until a single digit (1 through 9) is all that's left. For 7834 : 7 + 8 + 3 + 4 = 22, then 2 + 2 = 4 . That final digit is the number's check digit — its fingerprint for this trick. (Mathematicians call it the digital root .) Keep folding the digits until one is left Here's the shortcut that gives the trick its name. Before you add a single digit, throw away every 9 , and throw away any group of digits that adds up to 9 . They change nothing, so they can go. In 4592 , cast out the 9, then cast out the 4-and-5 pair (4 + 5 = 9), and you are left holding just the 2 — the check digit, with almost no arithmetic. Adding all four digits gives the same 2 — casting out the nines just gets there faster: 3. Check a Sum Without Redoing It Now the payoff. To check 234 + 567 = 801 , shrink each piece: 234 casts to 9 , 567 casts to 9 , and 9 + 9 folds back to 9 . Shrink the answer too: 801 casts to 9 . They match, so the sum passes. If someone instead claims 802 , that answer casts to 1 — and 1 does not match the required 9, so the answer is wrong before you ever re-add it. (The same check works for subtraction and multiplication, too.) The parts' check digits must fold to the answer's check digit

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