Long division in simple words
Long division splits a big number into equal groups, one digit at a time from the left. The number being split is the dividend, the number you divide by is the divisor, and the answer is the quotient.
Each round has four small jobs: divide, multiply, take away, then bring down the next digit.
For example, 12345 ÷ 67: 67 does not fit into 1 or 12, so start with 123. 67 goes into 123 once, and 123 − 67 = 56. Bring down 4 to make 564. 67 goes in 8 times, and 564 − 536 = 28. Bring down 5 to make 285. 67 goes in 4 times, and 285 − 268 = 17.
So 12345 ÷ 67 = 184 with 17 left over (the remainder). Check: 67 × 184 + 17 = 12345.
Formula and method
For positive divisor b, choose q and r so a = bq + r with 0 ≤ r < b. This is Euclidean integer division.
Worked example
12,345 divided by 67 gives quotient 184 and remainder 17, since 67 × 184 + 17 = 12,345.
Decimal quotients and repeating decimals
To divide by a decimal, move the decimal point the same number of places in both numbers until they are whole: 12.5 ÷ 0.25 becomes 1250 ÷ 25 = 50. Then divide as usual, bringing down a zero after each remainder to produce the next decimal digit.
If a remainder repeats, the digits repeat from that point on. For 1 ÷ 6 the remainders go 4, 4, 4…, so 1 ÷ 6 = 0.1666… = 0.1(6). For 1 ÷ 7 the block 142857 repeats. A fraction in lowest terms terminates only when its denominator has no prime factors other than 2 and 5.