The short answer
Long division breaks a division problem into repeated steps: divide, multiply, subtract, bring down. For 1,234 ÷ 12: divide 12 into the first digits, multiply the divisor by that digit, subtract, bring down the next digit, and repeat. The digits above the line form the quotient; whatever's left after the last subtraction is the remainder.
Key takeaways
- Every step follows the same four-move cycle — divide, multiply, subtract, bring down — repeated until there are no more digits to bring down.
- A remainder of 0 means the division is exact; a nonzero remainder means the answer either continues as a decimal or is best left as "quotient remainder R."
- Whether a decimal terminates or repeats forever depends only on the divisor's prime factors in lowest terms — 2s and 5s terminate, any other prime factor repeats.
- Dividing by a decimal? Shift the decimal point the same number of places in both dividend and divisor until the divisor is a whole number, then divide normally.
The four-step long division cycle
| Step | Action |
|---|---|
| 1. Divide | How many times does the divisor fit into the current digits? |
| 2. Multiply | Multiply the divisor by that digit |
| 3. Subtract | Subtract to find what's left over |
| 4. Bring down | Bring down the next digit and repeat |
Worked example: 1,234 ÷ 12
12 into 12 (first two digits) = 1, remainder 0
Bring down 3 → 03; 12 into 3 = 0, remainder 3
Bring down 4 → 34; 12 into 34 = 2, remainder 10
Quotient: 102, remainder 10
So 1,234 ÷ 12 = 102 remainder 10, or 102.8333... as a repeating decimal. Since 12 = 2² × 3, and 3 isn't a factor of 2 or 5, the decimal never terminates — it repeats the digit 3 forever.
Dividing by a decimal
12.5 ÷ 0.25
Shift both two places right (0.25 has 2 decimal places): 1250 ÷ 25
1250 ÷ 25 = 50
Multiplying both numbers by the same power of 10 doesn't change the answer — it just clears the decimal from the divisor so ordinary long division applies. This is the same principle as multiplying a fraction's numerator and denominator by the same number.
Common mistakes to avoid
- Forgetting to bring down a digit before continuing — this shifts every digit that follows and throws off the whole answer.
- Misplacing the decimal point in the quotient — it belongs directly above the decimal point in the dividend, not wherever feels natural.
- Reporting a remainder when a decimal answer was expected, or vice versa — check what form the question actually wants.
- Shifting the dividend's decimal point without shifting the divisor's by the same amount, which changes the value of the problem.
Related calculators
- Basic Calculator — get a quick decimal answer without the full step-by-step breakdown.
- Fraction Calculator — express the same division as an exact fraction instead of a decimal.
- Rounding Calculator — round a repeating or long decimal result to a usable number of places.
- Percentage Calculator — convert a division result directly into a percentage.