Long division is a method for dividing any two whole numbers by breaking a large problem into a series of smaller, manageable steps. If you can multiply single-digit numbers and subtract, you can do long division — it just takes practice with the pattern.
This guide walks through the method from scratch, builds up through increasingly challenging examples, and covers the edge cases that cause the most confusion: zeros in the quotient, remainders, and converting to decimals.
Use our long division calculator to check your work on any problem as you practice.
The Four Steps: Divide, Multiply, Subtract, Bring Down
Every long division problem uses the same four-step cycle, repeated until you run out of digits. A common mnemonic is "Does McDonald's Sell Burgers?" — the first letter of each word matches the first letter of each step.
- Divide: How many times does the divisor fit into the current working number?
- Multiply: Multiply the divisor by that quotient digit.
- Subtract: Subtract the product from the working number.
- Bring down: Bring down the next digit from the dividend to form a new working number.
Repeat until no digits remain. Whatever is left after the final subtraction is the remainder.
The Key Terms
Before diving in, here's the vocabulary:
- Dividend: The number being divided (the larger number)
- Divisor: The number you're dividing by
- Quotient: The answer (how many times the divisor fits)
- Remainder: What's left over when the divisor doesn't fit evenly
The relationship between them: (Quotient × Divisor) + Remainder = Dividend. This is how you check every answer.
Example 1: A Simple Start — 96 ÷ 4
This is a good first example because each step produces a clean result.
24
┌─────
4 │ 96
│ 8 (4 × 2 = 8)
│ ──
│ 16 (9 − 8 = 1, bring down 6)
│ 16 (4 × 4 = 16)
│ ──
│ 0 ← No remainder
Step by step:
- Divide: How many times does 4 go into 9? Twice (because 4 × 2 = 8, and 4 × 3 = 12 would be too large).
- Multiply: 4 × 2 = 8. Write 8 below the 9.
- Subtract: 9 − 8 = 1.
- Bring down: Bring down the 6 to make 16.
- Divide: How many times does 4 go into 16? Four times (4 × 4 = 16).
- Multiply: 4 × 4 = 16. Write 16 below.
- Subtract: 16 − 16 = 0. No more digits to bring down.
Answer: 96 ÷ 4 = 24
Check: 24 × 4 = 96 ✓
Example 2: With a Remainder — 157 ÷ 6
Most division problems don't come out evenly. That's normal — you just report the remainder.
26 R 1
┌───────
6 │ 157
│ 12 (6 × 2 = 12)
│ ───
│ 37 (15 − 12 = 3, bring down 7)
│ 36 (6 × 6 = 36)
│ ───
│ 1 ← Remainder
Step by step:
- 6 goes into 15 twice (6 × 2 = 12). Write 2 above the 5.
- Subtract: 15 − 12 = 3.
- Bring down the 7 to make 37.
- 6 goes into 37 six times (6 × 6 = 36). Write 6 above the 7.
- Subtract: 37 − 36 = 1. No more digits.
Answer: 157 ÷ 6 = 26 remainder 1
Check: (26 × 6) + 1 = 156 + 1 = 157 ✓
A key rule: the remainder must always be smaller than the divisor. If your remainder is 6 or larger (in this case), you underestimated the quotient digit.
Example 3: Multi-Digit Divisor — 4,738 ÷ 23
Dividing by a two-digit (or larger) number follows the exact same steps. The only difference is that estimating quotient digits takes a bit more thought.
205 R 23 → Wait, let me redo...
Actually, let's work through this carefully:
206 R 0
┌──────────
23 │ 4738
│ 46 (23 × 2 = 46)
│ ───
│ 13 (47 − 46 = 1, bring down 3)
│ 0 (23 × 0 = 0)
│ ───
│ 138 (13 − 0 = 13, bring down 8)
│ 138 (23 × 6 = 138)
│ ───
│ 0 ← No remainder
Step by step:
- 23 doesn't fit into 4, so look at 47. 23 × 2 = 46. Write 2 above the 7.
- Subtract: 47 − 46 = 1.
- Bring down the 3 to make 13. 23 doesn't fit into 13 (23 > 13), so write 0 in the quotient.
- Bring down the 8 to make 138. 23 × 6 = 138. Write 6 above the 8.
- Subtract: 138 − 138 = 0. Done.
Answer: 4,738 ÷ 23 = 206
Check: 206 × 23 = 4,738 ✓
The Estimation Trick for Multi-Digit Divisors
When dividing by a two-digit number, round the divisor to the nearest ten to estimate:
- Dividing by 23? Think "about 20." How many times does 20 go into 138? About 7. Try 23 × 7 = 161 (too big). Try 23 × 6 = 138 (perfect).
- Dividing by 47? Think "about 50." Adjust down if needed.
This trial-and-error is normal. Even experienced mathematicians estimate and adjust.
Example 4: Zeros in the Quotient — 3,024 ÷ 6
Zeros in the middle of the quotient trip up more students than anything else. The rule: if the divisor doesn't fit into the working number, write 0 and bring down the next digit.
504
┌───────
6 │ 3024
│ 30 (6 × 5 = 30)
│ ───
│ 02 (30 − 30 = 0, bring down 2)
│ 0 (6 × 0 = 0; 6 doesn't fit into 2)
│ ───
│ 24 (2 − 0 = 2, bring down 4)
│ 24 (6 × 4 = 24)
│ ──
│ 0 ← No remainder
Answer: 3,024 ÷ 6 = 504
If you forget to write the 0, you get 54 instead of 504 — a completely wrong answer. Every digit position in the dividend must produce a digit in the quotient (once you've started writing quotient digits).
Example 5: When the Dividend Is Smaller — 7 ÷ 12
When the dividend is smaller than the divisor, the answer is straightforward:
- Quotient: 0
- Remainder: 7 (the entire dividend)
As a fraction: 7/12. As a decimal: approximately 0.583.
No long division steps are needed — the divisor simply doesn't fit.
How to Convert Remainders to Decimals
Sometimes you need a decimal answer rather than a remainder. The process continues the same four steps — you just add a decimal point and keep dividing with zeros.
Example: 157 ÷ 6 as a decimal
We already know 157 ÷ 6 = 26 R 1. To continue:
- Write the decimal point in the quotient after 26.
- Add a 0 after the remainder 1, making it 10.
- 6 goes into 10 once (6 × 1 = 6). Subtract: 10 − 6 = 4.
- Add another 0: 40. 6 goes into 40 six times (6 × 6 = 36). Subtract: 40 − 36 = 4.
- The pattern 4 → 40 → 4 repeats forever.
Answer: 157 ÷ 6 = 26.1666... (the 6 repeats)
This is written as 26.1̄6̄ or rounded to 26.167 for practical purposes.
When to stop?
- If the remainder becomes 0, you're done — the decimal terminates.
- If you see the same remainder repeat, the decimal repeats. You can stop and note the pattern.
- For homework or practical use, 2–4 decimal places is usually enough.
Common Mistakes and How to Fix Them
These are the errors that cause the most wrong answers in long division:
1. Skipping the zero in the quotient
The mistake: When the divisor doesn't fit into the current working number, moving on without writing 0.
Why it matters: In 3,024 ÷ 6, skipping the zero gives 54 instead of 504.
The fix: Every time you bring down a digit (after the first quotient digit), you must write something above it — either a digit 1–9 or a 0.
2. Subtraction errors
The mistake: Getting the subtraction wrong, especially when borrowing is needed.
Why it matters: One wrong subtraction makes every following step wrong too.
The fix: Double-check each subtraction before moving on. If your final check (Quotient × Divisor + Remainder = Dividend) fails, a subtraction error is the most likely cause.
3. Overestimating or underestimating the quotient digit
The mistake: Guessing that 23 goes into 138 seven times (23 × 7 = 161, too large) or five times (23 × 5 = 115, leaving 23 — which means you could fit one more).
How to tell:
- Too high: The product is larger than the working number. Reduce by 1.
- Too low: The remainder after subtracting is ≥ the divisor. Increase by 1.
The fix: After multiplying and subtracting, check that the remainder is between 0 and (divisor − 1).
4. Not aligning digits properly
The mistake: Writing quotient digits in the wrong position, leading to a misplaced answer.
The fix: Each quotient digit goes directly above the dividend digit that was last brought down. Use grid paper or draw vertical lines if alignment is tricky.
5. Forgetting to bring down exactly one digit
The mistake: Bringing down two digits at once or none at all.
The fix: Always bring down exactly one digit from the dividend per cycle.
Mental Shortcuts and Divisibility Rules
Before starting long division, check whether the problem has a simpler path:
| Divisor | Quick test |
|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8) |
| 3 | Digit sum is divisible by 3 |
| 4 | Last two digits form a number divisible by 4 |
| 5 | Last digit is 0 or 5 |
| 6 | Divisible by both 2 and 3 |
| 8 | Last three digits form a number divisible by 8 |
| 9 | Digit sum is divisible by 9 |
| 10 | Last digit is 0 |
Example: Is 4,738 divisible by 3? Digit sum: 4 + 7 + 3 + 8 = 22. 22 is not divisible by 3, so 4,738 ÷ 3 will have a remainder.
These checks don't replace long division, but they help you predict whether the result will be clean or have a remainder.
Practice Problems
Try these on your own, then check your answers. Work through all four steps for each problem.
Beginner:
- 84 ÷ 7
- 156 ÷ 4
- 245 ÷ 5
Intermediate: 4. 1,347 ÷ 9 5. 5,672 ÷ 8 6. 837 ÷ 12
Challenging: 7. 9,108 ÷ 36 8. 12,005 ÷ 25 9. 7,777 ÷ 77
Answers
- 84 ÷ 7 = 12 (Check: 12 × 7 = 84)
- 156 ÷ 4 = 39 (Check: 39 × 4 = 156)
- 245 ÷ 5 = 49 (Check: 49 × 5 = 245)
- 1,347 ÷ 9 = 149 R 6 (Check: 149 × 9 + 6 = 1,341 + 6 = 1,347)
- 5,672 ÷ 8 = 709 (Check: 709 × 8 = 5,672)
- 837 ÷ 12 = 69 R 9 (Check: 69 × 12 + 9 = 828 + 9 = 837)
- 9,108 ÷ 36 = 253 (Check: 253 × 36 = 9,108)
- 12,005 ÷ 25 = 480 R 5 (Check: 480 × 25 + 5 = 12,000 + 5 = 12,005)
- 7,777 ÷ 77 = 101 (Check: 101 × 77 = 7,777)
Notice problem 5: 5,672 ÷ 8 = 709. That zero in the quotient is exactly the trap described earlier. Did you catch it?
Verify any of these instantly with the long division calculator.
Long Division vs. Short Division
Short division is the same process but condensed: you do the multiplication and subtraction mentally and only write the quotient. It's faster once you're comfortable, but harder to learn from because you can't see where errors happen.
| Long division | Short division | |
|---|---|---|
| Steps shown | All written out | Done mentally |
| Best for | Learning, large divisors | Quick calculations, small divisors |
| Error detection | Easy to trace | Hard to find mistakes |
| When to use | Multi-digit divisors, studying | Single-digit divisors you know well |
Start with long division until the process is automatic, then transition to short division for simple problems.
Why Long Division Still Matters
Even with calculators available, long division builds skills that transfer:
- Number sense: Understanding why 4,738 ÷ 23 ≈ 200 (not 20 or 2,000) helps you catch calculator typos and estimate in daily life.
- Algebra preparation: Polynomial long division uses the exact same four-step process. Students who are comfortable with numerical long division learn polynomial division faster.
- Fraction understanding: Long division is literally what a fraction means: 3/4 is the same as 3 ÷ 4 = 0.75. Understanding this connection is foundational.
- Systematic problem-solving: The divide-multiply-subtract-bring-down cycle is a pattern: break a hard problem into repeating simple steps. That approach applies far beyond math.
Frequently Asked Questions
What is long division? Long division is a standard written method for dividing numbers. It breaks a division problem into a series of four repeating steps — divide, multiply, subtract, bring down — producing a quotient and remainder. It works for any two whole numbers, regardless of size.
How do you do long division step by step? Start by seeing how many times the divisor fits into the first digit(s) of the dividend. Write that digit in the quotient, multiply it by the divisor, subtract the result, then bring down the next digit. Repeat until no digits remain. The final subtraction result is your remainder.
What if the divisor is bigger than the first digit? Look at more digits together. For example, in 4,738 ÷ 23, the divisor 23 doesn't fit into 4, so you look at 47 instead. 23 fits into 47 twice.
How do you know when you're done? When you've brought down and processed every digit of the dividend. If you want a decimal answer, you can continue by adding zeros and extending the division.
What does "remainder" mean? The remainder is the amount left over after the divisor has been subtracted as many whole times as possible. In 157 ÷ 6 = 26 R 1, the 1 means that after dividing 157 into groups of 6, one unit is left that doesn't fill another complete group.
How do I check my long division answer? Multiply the quotient by the divisor and add the remainder. The result must equal the original dividend: (Quotient × Divisor) + Remainder = Dividend.
Can long division give a decimal answer? Yes. After reaching the remainder, add a decimal point to the quotient and continue dividing by appending zeros. For instance, 157 ÷ 6 = 26.1666... as a decimal.
What is a repeating decimal? When the same remainder appears again during decimal division, the decimal digits start repeating in a cycle. For example, 1 ÷ 3 = 0.333... (the 3 repeats forever). This happens whenever the division doesn't terminate.
Why do I need to write a zero in the quotient sometimes? When the divisor doesn't fit into the current working number, you must write 0 to hold that digit's place in the quotient. Omitting the zero shifts all following digits left, giving a wrong answer (e.g., 54 instead of 504).
Is long division the same as short division? Short division uses the same logic but does the multiplication and subtraction mentally. Long division writes every step out, which makes it easier to learn and easier to find mistakes.
What is the hardest part of long division? For most students, estimating quotient digits with multi-digit divisors and remembering to write zeros when the divisor doesn't fit. Both get easier with practice.
When do students typically learn long division? Long division is usually introduced in 4th or 5th grade (ages 9–11) in most curricula, starting with single-digit divisors and progressing to multi-digit divisors.
Can you do long division with negative numbers? Divide the absolute values first, then apply the sign rule: if the dividend and divisor have the same sign, the quotient is positive. If they have different signs, the quotient is negative.
How is long division related to fractions? Every fraction is a division problem. The fraction 3/4 means 3 ÷ 4. Performing the long division gives 0.75. This is why long division is foundational for understanding fractions and decimals.