Casting out nines
A quick check for long sums and products that bookkeepers used for centuries before calculators.
Replace every number with the sum of its digits, and repeat until a single digit is left. That digit is the number's remainder after division by 9 (with 9 standing for 0).
4,728 → 4+7+2+8 = 21 → 3 + 3,195 → 3+1+9+5 = 18 → 9 (≡ 0) = 7,923 → 7+9+2+3 = 21 → 3 3 + 0 = 3 ✓
It works for multiplication too: reduce both factors, multiply the reductions, reduce again, and compare with the reduced answer.
What it misses
The check is blind to any mistake that changes the answer by a multiple of 9. The most common one is swapping two digits: 7,923 and 7,293 have the same digit sum. A dropped zero also slips through. For those, casting out elevens (alternating sums of digits) is the usual companion check.