Learn how the traditional long division method works through step-by-step examples with one- and two-digit divisors. Understand each stage of the process, including quotient placement, bringing down digits, handling remainders, and checking division answers.
What Is Long Division?
Long division breaks a large division problem into smaller calculations. Students work through these calculations in a set order to divide multi-digit numbers by one-digit or two-digit divisors.
The method follows five steps:
- Divide
- Multiply
- Subtract
- Bring Down
- Repeat the Process
Each step builds on the previous one. Students divide, multiply, and subtract before bringing down the next digit of the dividend. They continue until every digit has been used. The quotient digits are written above the dividend, with each digit in its correct place-value position.
Let’s work through 458 ÷ 3 to see how the steps fit together. The aim is to find how many times 3 goes into 458 and what remains.
How to Divide Using Long Division
Step 1: Divide
First, set up the problem 458 ÷ 3.
- Draw the long division symbol.
- Write the divisor, 3, outside the symbol on the left.
- Write the dividend, 458, inside the symbol.
Divide the first digit.
Start with the first digit of the dividend, 4. Determine how many times 3 goes into 4 without exceeding it.
Since:
3 × 1 = 3
3 goes into 4 once.
Write the first quotient digit.
Place 1 above the 4. This is the first digit of the quotient.
Keep the quotient digits aligned with the corresponding digits of the dividend. Their positions determine the place value of each digit in the answer.
Step 2: Multiply
Multiply the divisor by the quotient digit you just wrote:
3 × 1 = 3
Write 3 beneath the 4, keeping the digits aligned. This gives you the amount to subtract from 4.
Step 3: Subtract
Subtract the product from the number above it.
- Draw a line beneath the product.
- Subtract 3 from 4.
- Write the result beneath the line.
4 − 3 = 1
The difference is 1.
Step 4: Bring Down
The divisor, 3, cannot go into 1 without exceeding it. Bring down the next digit of the dividend, 5, and place it beside the 1.
You now have 15 to divide.
Step 5: Repeat the Process
Return to the first three steps: divide, multiply, and subtract.
Divide 15 by 3.
Since:
3 × 5 = 15
3 goes into 15 five times. Write 5 above the next digit of the dividend.
Multiply and subtract.
Multiply:
3 × 5 = 15
Write 15 beneath 15, then subtract:
15 − 15 = 0
Bring down the next digit.
Bring down the final digit, 8. It joins the 0 to make 08, which is simply 8.
Divide 8 by 3.
3 goes into 8 twice because:
3 × 2 = 6
Write 2 above the final digit of the dividend.
Multiply:
3 × 2 = 6
Subtract:
8 − 6 = 2
There are no more digits to bring down, so 2 is the final remainder.
The digits above the division symbol give the quotient, 152. The final remainder is 2.
Therefore:
458 ÷ 3 = 152 R 2
The answer is 152 remainder 2.
For more practice with one-digit divisors, the 1-Digit Divisors Long Division Worksheets Bundle provides problems with different dividend sizes. You can also browse the Long Division Worksheets collection for individual worksheet sets and other practice options.
How Does Long Division Work with Two-Digit Divisors?
The same five steps apply when dividing by a two-digit divisor. However, you may need to use the first two or three digits of the dividend to find a number large enough to divide.
Consider 672 ÷ 24. Since 24 is greater than the first digit, 6, start with the first two digits, 67.
Step 1: Divide
Set up the problem with 24 outside the division symbol and 672 inside.
Since 24 cannot go into 6, consider 67.
Find the largest number of times 24 goes into 67 without exceeding it.
Since:
24 × 2 = 48
and
24 × 3 = 72
24 goes into 67 twice. Write 2 above the 7, in the tens position of the quotient.
Step 2: Multiply
Multiply the divisor by the quotient digit:
24 × 2 = 48
Write 48 beneath 67, keeping the digits aligned.
Step 3: Subtract
Subtract the product from 67:
67 − 48 = 19
Write 19 beneath the subtraction line.
Step 4: Bring Down
Bring down the next digit of the dividend, 2, and place it beside 19.
The new number is 192.
Step 5: Repeat the Process
Find how many times 24 goes into 192.
Since:
24 × 8 = 192
24 goes into 192 exactly eight times. Write 8 above the final digit of the dividend.
Multiply:
24 × 8 = 192
Subtract:
192 − 192 = 0
There are no more digits to bring down, and nothing remains.
The quotient digits above the division symbol give 28.
The first digit, 2, represents 2 tens. The second digit, 8, represents 8 ones.
Therefore:
672 ÷ 24 = 28
Check the answer:
24 × 28 = 672
The product matches the original dividend, so the answer is correct.
For more practice with two-digit divisors, the 2-Digit Divisors Long Division Worksheets Bundle provides problems with different dividend sizes. You can also browse the Long Division Worksheets collection for individual worksheet sets and other practice options.
Understanding Remainders in Long Division
A remainder is the amount left after dividing as many complete groups as possible.
In 458 ÷ 3, the final subtraction is:
8 − 6 = 2
Since there are no more digits to bring down, 2 is the remainder.
The remainder must always be smaller than the divisor. If it is equal to or greater than the divisor, another complete group can still be formed, so the calculation is not finished.
Write the remainder after the quotient, using the letter R to separate the two.
For example:
458 ÷ 3 = 152 R 2
Why Does Long Division Work?
Each quotient digit tells you how many times the divisor goes into the number being divided at that stage. Multiplying the divisor by the quotient digit gives the amount removed from that number.
Subtracting leaves the difference. Bringing down the next digit joins it to the remainder, creating the next number to divide.
In 458 ÷ 3, the process works as follows:
- 3 goes into 4 once, leaving 1.
- Bring down 5 to make 15.
- 3 goes into 15 five times, leaving 0.
- Bring down 8.
- 3 goes into 8 twice, leaving 2.
The quotient digits are 1, 5, and 2, giving 152. The final remainder is 2.
The method also follows the relationship between division and multiplication:
Dividend = (Divisor × Quotient) + Remainder
For this example:
458 = (3 × 152) + 2
458 = 456 + 2
The result is the original dividend, confirming the quotient and remainder.
Common Errors in Long Division
Choosing the Wrong Number to Divide
When the divisor is greater than the first digit of the dividend, students need to consider the first two digits instead.
For example, in 672 ÷ 24, 24 cannot go into 6, so the calculation starts with 67.
Starting with the wrong number can lead to an incorrect quotient.
Misplacing Quotient Digits
Each quotient digit must sit above the appropriate digit of the dividend. This keeps the place values correctly aligned.
In 458 ÷ 3, the quotient digits 1, 5, and 2 must appear above the corresponding digits 4, 5, and 8.
Choosing a Multiple That Is Too Large
Choose a multiple of the divisor that does not exceed the number being divided.
For example, 24 goes into 67 twice because:
24 × 2 = 48
But:
24 × 3 = 72
Since 72 is greater than 67, three is too many. Check the multiplication before writing the product and subtracting.
Making Subtraction Errors
Subtract the product from the current number being divided, not from the original dividend.
For example, after multiplying 3 by 5 in 458 ÷ 3, subtract 15 from 15:
15 − 15 = 0
Check each subtraction before bringing down the next digit.
Bringing Down the Wrong Digit
Bring down only the next unused digit of the dividend after completing the subtraction.
Skipping a digit or bringing down more than one digit changes the number being divided.
Forgetting to Repeat the Steps
After bringing down the next digit, return to dividing, multiplying, and subtracting. Continue until every digit of the dividend has been used.
Recording an Incorrect Remainder
The final remainder must be smaller than the divisor. If it is equal to or greater than the divisor, the calculation is not complete.
In 458 ÷ 3, the remainder is 2, which is smaller than 3.
How to Check a Long Division Answer
To check a division answer, multiply the quotient by the divisor and add the remainder.
Use this relationship:
Dividend = (Divisor × Quotient) + Remainder
For 458 ÷ 3 = 152 R 2:
458 = (3 × 152) + 2
458 = 456 + 2
458 = 458
The calculation returns the original dividend, confirming that the answer is correct.
When there is no remainder, multiply the divisor by the quotient. The product should equal the original dividend.
Long Division and Other Division Methods
Long division is one of several methods students can use to divide multi-digit numbers. Other approaches include partial quotients, area model division, and the Box Method.
Each method organises the calculation differently. Traditional long division follows a fixed sequence of dividing, multiplying, subtracting, and bringing down the next digit. The Box Method uses a rectangular layout and brings up the remainder into the next column instead.
To compare the different approaches, read Division Strategies for Grade 4 and Grade 5. You can also read the individual guides to Area Model Division, Partial Quotients Division, and Box Method Division.
Final Check: Understanding Long Division
Students should be able to explain each step, why the next digit is brought down beside the remainder, and how to position quotient digits above the dividend. They should also be able to divide using one-digit and two-digit divisors, identify the final remainder, and check their answers.
Long division gives students a consistent way to solve multi-digit division problems. Once they understand how dividing, multiplying, subtracting, and bringing down work together, they can carry out the calculation and explain how they found the answer.
To compare long division with other approaches, read Division Strategies for Grade 4 and Grade 5.
