Arithmetic
How to Do Long Division
Long division splits a big division into small steps, one digit at a time: divide, multiply, subtract, bring down, and repeat until no digits are left. This guide works through full examples, including answers with remainders, zeros in the quotient, two-digit divisors and decimal answers, explains why the method works, and lists the mistakes that cause most wrong answers.
How to do long division: the four steps
Long division breaks a big division into a chain of small ones, one digit of the dividend at a time. At each digit you repeat the same four steps: divide, multiply, subtract, bring down. When there are no digits left to bring down, whatever is left over is the remainder.
First, the names. In 852 ÷ 4, 852 is the dividend (the number being divided), 4 is the divisor (the number you divide by), and the answer is the quotient. The dividend goes inside the division bracket, the divisor outside to the left, and the quotient is built on top, one digit at a time.
- Divide. Look at the first digit (or first few digits) of the dividend. How many times does the divisor fit into it? Write that number on top.
- Multiply. Multiply the digit you just wrote by the divisor. Write the result under the digits you were dividing.
- Subtract. Subtract to find what is left over. It must be smaller than the divisor.
- Bring down. Bring the next digit of the dividend down next to the remainder. That makes the new number to divide. Go back to step 1.
A common memory aid is “Does McDonald’s Sell Burgers?” for Divide, Multiply, Subtract, Bring down. The long division calculator lays out every one of these steps for any numbers you enter, so you can check your own working line by line.
A worked example: 852 ÷ 4
Example: 852 ÷ 4
Start with the first digit, 8. Four goes into 8 exactly 2 times. Write 2 above the 8, multiply 2 × 4 = 8, subtract to get 0.
Bring down the 5. Four goes into 5 once. Write 1 above the 5, multiply 1 × 4 = 4, subtract to get 1.
Bring down the 2, making 12. Four goes into 12 exactly 3 times. Write 3 above the 2, multiply 3 × 4 = 12, subtract to get 0. No digits are left, and the remainder is 0.
So 852 ÷ 4 = 213. Check: 213 × 4 = 852.
Notice that each digit of the quotient sits directly above the last digit of the number it came from. Keeping the columns lined up is most of the battle; the arithmetic at each step only needs the times tables. If any of those facts are shaky, a multiplication chart beside you while you practise removes the guesswork.
Why long division works
Long division is place value in disguise. 852 is 8 hundreds, 5 tens and 2 ones. Sharing it among 4 one place at a time looks like this:
“Bring down” is the step where leftover tens get traded for ones (or leftover hundreds for tens). The remainder from one place is worth ten times as much in the next place, which is why it slides down and sits in front of the next digit. The guide to place value covers the columns this method depends on.
Long division with remainders
Most divisions do not come out evenly. When the last subtraction leaves a number bigger than 0, that number is the remainder.
Example: 745 ÷ 6
Six goes into 7 once, remainder 1. Bring down the 4 to make 14; six goes into 14 twice, remainder 2. Bring down the 5 to make 25; six goes into 25 four times, remainder 1. Nothing is left to bring down, so the answer is 124 remainder 1.
A remainder can be written three ways, and which one you want depends on the question:
| Form | 745 ÷ 6 | When to use it |
|---|---|---|
| Remainder | 124 r 1 | Sharing whole objects that cannot be split |
| Mixed number | 124 1/6 | When an exact answer is needed; the remainder goes over the divisor |
| Decimal | ≈ 124.17 (rounded) | Money, measurements, anything shown on a calculator |
To check any answer with a remainder, multiply the quotient by the divisor and add the remainder. You should get the dividend back: 124 × 6 + 1 = 745. The remainder calculator gives the quotient and remainder together, and the mixed number calculator handles the fraction form.
Zeros in the quotient
The step most often skipped is the one where the divisor does not fit at all. When you bring down a digit and the number is still smaller than the divisor, you must write a 0 in the quotient, then bring down the next digit.
Example: 1,236 ÷ 6
Six does not go into 1, so start with 12. Six goes into 12 twice, remainder 0. Bring down the 3. Six does not go into 3, so write 0 on top. Bring down the 6, making 36. Six goes into 36 six times.
The answer is 206, not 26. Check: 206 × 6 = 1,236. Leaving out that zero shifts every later digit into the wrong place value, and the answer comes out ten times too small.
When the first digit of the dividend is smaller than the divisor, as with the 1 here, you can leave the space above it blank rather than writing a leading zero. A zero in the middle or at the end of the quotient must always be written.
Long division with two-digit divisors
The four steps do not change when the divisor has two digits. What changes is the divide step: you no longer know the answer from the times tables, so you estimate, then adjust.
Example: 4,165 ÷ 17
Seventeen does not go into 4, so start with 41. Round 17 to 20 to estimate: 20 goes into 41 about 2 times. Check: 2 × 17 = 34, which fits. Subtract to get 7.
Bring down the 6, making 76. The estimate says 20 goes into 76 about 3 times, but 3 × 17 = 51 leaves 25, which is bigger than 17. The estimate was too low, so try 4: 4 × 17 = 68, leaving 8. Write 4.
Bring down the 5, making 85. The estimate says 4, but 5 × 17 = 85 fits exactly. Write 5.
4,165 ÷ 17 = 245. Check: 245 × 17 = 4,165.
Rounding the divisor gives a starting guess, not the answer. If the leftover after subtracting is as big as the divisor or bigger, raise the digit by one. If the product is bigger than the number you are dividing, lower it by one. Writing out a quick list of the divisor’s multiples first (17, 34, 51, 68, 85, 102…) removes the guessing altogether. The multiplication calculator helps when the divisor is large.
Carrying on past the decimal point
Instead of stopping with a remainder, you can keep dividing to get a decimal answer. Put a decimal point after the dividend, add zeros, and continue bringing them down. Put a decimal point in the quotient directly above the one in the dividend.
Example: 25 ÷ 4
Four goes into 25 six times, remainder 1. Bring down a 0 to make 10; four goes into 10 twice, remainder 2. Bring down another 0 to make 20; four goes into 20 five times, remainder 0. The answer is 6.25.
Some divisions never finish. 1 ÷ 3 gives a remainder of 1 at every step, so the 3 repeats forever: 0.333… When you see the same remainder come back, the digits will repeat from there, and you can stop and round. The fraction to decimal calculator marks repeating decimals for you, and the guide to rounding to decimal places covers where to cut them off.
When the divisor is a decimal, you move the decimal point first and then divide as normal. That is a separate method, explained in how to divide decimals.
Divisibility rules that speed up long division
Before you start, a quick divisibility check tells you whether to expect a remainder. These rules are standard and work for any whole number.
| Divisible by | Rule | Example |
|---|---|---|
| 2 | Last digit is even | 852 ends in 2: yes |
| 3 | Digit sum is divisible by 3 | 7,825: 7 + 8 + 2 + 5 = 22: no |
| 4 | Last two digits divisible by 4 | 852: 52 ÷ 4 = 13: yes |
| 5 | Last digit is 0 or 5 | 745 ends in 5: yes |
| 6 | Divisible by both 2 and 3 | 1,236: even, digit sum 12: yes |
| 9 | Digit sum is divisible by 9 | 4,165: digit sum 16: no |
| 10 | Last digit is 0 | 4,165: no |
The factor calculator lists every number that divides evenly into a given number, which is the same question asked the other way round.
Common long division mistakes
- Forgetting a zero in the quotient. If a brought-down number is smaller than the divisor, write 0 on top before bringing down the next digit. Skipping it is the most common error, and it makes the answer ten times too small.
- Letting the columns drift. Each quotient digit goes directly above the last digit of the number it divided. Squared paper helps.
- A remainder that is too big. If the leftover is equal to or larger than the divisor, the quotient digit was too small.
- Bringing down two digits at once. Bring down exactly one digit per cycle, and write a quotient digit for every one.
- Subtraction slips. Most wrong answers come from a borrowing error in the subtract step, not from the division itself.
- Not checking. Multiply the quotient by the divisor and add the remainder. If you do not get the dividend back, find the step where it went wrong.
For a fast answer without the working, the division calculator gives the result and remainder directly.
Frequently asked questions
What are the steps of long division?
Divide, multiply, subtract, bring down, and repeat. Divide the current number by the divisor and write the digit on top, multiply that digit by the divisor, subtract the product, then bring down the next digit of the dividend. When no digits remain, the last leftover is the remainder. The memory aid is “Does McDonald’s Sell Burgers?”
How do you do long division with remainders?
Work the four steps until no digits are left to bring down. The final leftover is the remainder, and it must be smaller than the divisor. For 745 ÷ 6 the answer is 124 remainder 1, which can also be written as 124 1/6 or about 124.17. Check with 124 × 6 + 1 = 745.
What is 852 divided by 4 using long division?
852 ÷ 4 = 213 with no remainder. Four goes into 8 twice, into 5 once with 1 left over, and into the resulting 12 three times. Multiplying back, 213 × 4 = 852, confirms the answer. Since 852 is even and its last two digits, 52, divide evenly by 4, you could tell before starting that there would be no remainder.
When do you put a zero in the quotient?
Whenever you bring down a digit and the number is still smaller than the divisor. Write 0 in the quotient above that digit, then bring down the next one. In 1,236 ÷ 6, after 12 comes 3, which 6 cannot divide, so a 0 goes on top and the answer is 206, not 26.
How do you do long division with two-digit divisors?
Use the same four steps, but estimate each quotient digit by rounding the divisor. For 4,165 ÷ 17, round 17 to 20 to get first guesses, then check each by multiplying. If the leftover is 17 or more, raise the digit; if the product is too big, lower it. The answer is 245.
How do you check a long division answer?
Multiply the quotient by the divisor, then add the remainder. The result must equal the dividend. For 745 ÷ 6 = 124 r 1, the check is 124 × 6 + 1 = 744 + 1 = 745. If the numbers do not match, recheck each subtraction, which is where most errors happen.
How do you get a decimal answer from long division?
When you run out of digits, put a decimal point after the dividend and in the quotient directly above it, then bring down zeros and keep dividing. For 25 ÷ 4, the remainder of 1 becomes 10, then 20, giving 6.25. If a remainder repeats, the decimal repeats too, as in 1 ÷ 3 = 0.333…
What is the difference between long division and short division?
Both follow the same steps. Long division writes out every multiply and subtract line under the dividend; short division does those in your head and writes only the small carried remainders. Long division is easier to follow and check, especially with two-digit divisors, while short division is quicker once single-digit facts are automatic.
Calculators for this
Every one of these shows the working, so you can check the method as well as the answer.
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