Casting Out Nines

Technique: Verify a product without recomputing it

If A × B = C, then the digit-sum of A times the digit-sum of B equals the digit-sum of C, all reduced mod 9. If the mod-9 check fails, the equation is wrong. Type 1 if the equation could be valid, 0 if mod-9 says it's wrong.

Worked example

73 × 84 = 6132?

• Digit-sum of 73: 7+3 = 10 → 1+0 = 1 (mod 9)
• Digit-sum of 84: 8+4 = 12 → 1+2 = 3 (mod 9)
• Expected: 1×3 = 3 (mod 9)
• Digit-sum of 6132: 6+1+3+2 = 12 → 3 (mod 9) — matches! Answer: 1 (could be valid)

Now 73 × 84 = 6135?

• Same expected: 3 (mod 9)
• Digit-sum of 6135: 6+1+3+5 = 15 → 1+5 = 6 (mod 9) — doesn't match. Answer: 0

Why does this work?

Because 10 ≡ 1 (mod 9), every digit contributes its face value to the number mod 9. So n ≡ digit-sum(n) (mod 9). Multiplication respects mod 9, so any arithmetic error that shifts the answer by anything other than a multiple of 9 will leave a different digit-sum. The check never gives a false negative — if mod 9 fails, the equation is definitely wrong. (Caveat: errors that happen to be multiples of 9 will sneak through. This trainer never presents those, so trust the rule.)

Every question has a 30 second shot clock.