School says 17 divided by 5 is “3 remainder 2”, and the remainder sounds like something left over from a job not quite finished. It is half of the answer. Division asks two questions at once, how many each and how many left, and both answers are exact.
On the table: share 17 stones among 5 cups, one round at a time, until there are not enough left for another round. Each cup has 3; 2 are left. Check it: 5 × 3 + 2 = 17.
For the teacher: quot_rem_unique proves there is only one such pair. If b × q + r = b × q′ + r′ with both remainders below b, then the quotients are equal and so are the remainders. quot_rem_exist proves that the pair always exists, and that it is what the machine's / and % compute. The script checks every n below 200 against every divisor from 1 to 12 and finds exactly one pair each time COMPUTED. Because it is unique, the remainder is not a leftover to be apologised for. It is a fact about the number, as definite as the quotient.
Book XI showed that the digits of a numeral are exactly such remainders: division by ten, repeated. So every digit a learner writes is the second answer to a division.
Long division is carrying run backwards. Divide 2026 by 7 working from the left, and the running remainders are 2, 6, 6, 3. At each step, multiply the remainder by ten, bring down the next digit, and keep only what is left after sevens. The last running remainder, 3, is the remainder of 2026 ÷ 7 COMPUTED.
remH_correct proves this always works, for any divisor. The one-step fact underneath, rem_step, is that bringing a digit down onto the remainder or onto the whole number so far leaves the same remainder. The script checks all divisors from 1 to 30 on every number below 20,000 COMPUTED.
An old way to check a calculation is to add up the digits. A number and the sum of its digits leave the same remainder on division by nine. nines proves it, and the script checks every number below 100,000 COMPUTED. So a sum or product whose digit-sums disagree mod 9 is certainly wrong.
It has a blind spot, and the teacher should know it. Swapping two digits does not change the digit sum, so casting out nines never catches a transposition, the most common copying error. In 18,079 random numbers with two neighbouring digits swapped, nines caught none and the check by eleven caught all of them COMPUTED. A check that passes is not a proof of correctness. This series has other names for that: see Book XX, ch. 3 on the errors an instrument is built to miss.