Long Division
Divide multi-digit numbers by 1- and 2-digit divisors with the long-division algorithm: divide, multiply, subtract, bring down — and check with multiplication.
1 The long-division chant
Long division repeats four moves until every digit has been used: Divide, Multiply, Subtract, Bring down.
The moves
Divide: how many times does the divisor fit into the working number? Write that digit in the quotient. Multiply: quotient digit × divisor. Subtract: take it away from the working number. Bring down: drop the next digit beside the difference and repeat.
Estimate the quotient digit first
Before writing a digit, estimate with compatible numbers: for 365 ÷ 12, think 360 ÷ 12 = 30. If your estimate times the divisor overshoots, the digit is too big.
Every place needs a digit
Sometimes the divisor does not fit and the quotient digit is 0 — write it! 612 ÷ 3 = 204, not 24. A missing 0 makes the answer ten times too small.
2 See it: 156 ÷ 4, step by step
Follow the chant down the left column; each move is labeled on the right.
3 Worked examples
Chant it as you go: divide, multiply, subtract, bring down.
- Divide: 4 goes into 15 three times. Write 3 in the quotient.
- Multiply: 3 × 4 = 12. Subtract: 15 − 12 = 3.
- Bring down the 6 → working with 36.
- Divide: 36 ÷ 4 = 9. Multiply: 36. Subtract: 0. Done.
- Check: 39 × 4 = 156. Correct.
- Divide: 6 ÷ 3 = 2. Multiply, subtract → 0. Bring down 1.
- Divide: 1 ÷ 3 = 0. Write the 0! Multiply 0 × 3 = 0, subtract → 1.
- Bring down 2 → 12. 12 ÷ 3 = 4.
- Check: 204 × 3 = 612. Correct.
- 8 ÷ 7 = 1, multiply 7, subtract → 1. Bring down 4 → 14.
- 14 ÷ 7 = 2, multiply 14, subtract → 0. Bring down 7.
- 7 ÷ 7 = 1, multiply 7, subtract → 0.
- Check: 121 × 7 = 847. Correct.
- Estimate first: 360 ÷ 12 = 30, so expect about 30.
- Divide: 12 into 36 goes 3. Multiply 36, subtract → 0. Bring down 5.
- Divide: 5 ÷ 12 = 0. Multiply 0, subtract → 5. No more digits.
- Check: 30 × 12 + 5 = 365. Correct.
4 Common mistakes
Two traps catch almost everyone.
5 Key vocabulary
- Long division
- The written algorithm that divides digit by digit.
- Quotient digit
- One digit of the answer, found by a single divide step.
- Bring down
- Dropping the next dividend digit beside the difference to form the new working number.
- Compatible numbers
- Nearby easy numbers used for estimating, like 360 and 12.
- Remainder
- What is left after the last subtraction; always smaller than the divisor.
6 Quick check
Try these, then reveal the answers.
Key points to remember
- Chant the moves: divide, multiply, subtract, bring down.
- Estimate each quotient digit with compatible numbers before writing it.
- When the divisor does not fit, the quotient digit is 0 — write it.
- Check: quotient × divisor + remainder = dividend.
- The remainder is always smaller than the divisor.