101 LearnersLast updated on July 14th, 2025
Long Division
The method used for dividing large numbers is long division. Long division follows the same principle as regular division, where the dividend is divided by the divisor to yield the quotient, with a possible remainder. Now, let’s learn how to divide using the long division method.
What is the Long Division Method?
Long division is a way of dividing large numbers into groups or parts. It makes the division easier by breaking it into small steps. As in regular division, the parts of the long division method are the dividend, divisor, quotient, and remainder. Here, the large number is the dividend and the number which divides the dividend is the divisor. The result of the division is the quotient. The remainder is the number left after division, which is less than the divisor or zero.
What are the Components of the Long Division Method?
The long division method has four components: dividend, divisor, quotient, and remainder. In this section, let’s learn about the components of the long division method.
- Dividend: The number which is divided is the dividend, it is the largest number among all the four components.
- Divisor: The number that divides the dividend is the divisor; it is smaller than the dividend.
- Quotient: The result of dividing a number is the quotient. The quotient can be greater than, less than, or equal to the divisor, depending on the dividend (e.g., in 20 ÷ 5 = 4, the quotient 4 is less than the divisor 5).
- Remainder: The remainder is the leftover of the long division. When the remainder is zero, the dividend is a multiple of the divisor.
For example, when dividing 452 by 4, that is 452 ÷ 4 = 113 with remainder 0. Here, 452 is the dividend, 4 is the divisor, 113 is the quotient, and 0 is the remainder.
What are the steps for Long Division?
Now let’s learn how to do the long division method, follow these steps to perform the long division method.
Step 1: Check if the first digit(or first group of digits) of the dividend is greater than or equal to the divisor.
For example, dividing 27 by 2. Here, the dividend (27) is greater than the divisor (2).
Step 2: Now find a multiple of the divisor which is less than or equal to the dividend. Make sure the quotient is less than 9 in this step.
Here, 2 × 1 = 2, the first digit in the quotient is 2.
Step 3: Find the difference between the multiple of the divisor and the first digit of the dividend. Here the difference is 0 as 2 - 2 = 0. 27 / 2 = 13.5
Step 4: Bring down the next digit (7), forming 07. Divide 7 ÷ 2 = 3 remainder 1 (since 2 × 3 = 6, 7 – 6 = 1). The quotient digit is 3.
Step 5: Repeat steps 2 to 4 until the remainder is zero or less than the divisor.
There are two cases in the long division method based on the first digit of the dividend. Now, let’s learn about the cases in detail.
Case 1: When the first digit of the dividend is greater than or equal to the divisor
Now let’s learn how to divide a number when the first digit of the dividend is greater than or equal to the divisor. For example, divide the 252 by 2
Step 1: Check if the dividend is less than or equal to the divisor from the left-hand side. Here the first digit is 2 and it is equal to the divisor.
Step 2: As 2 × 1 = 2, the quotient is 1.
Step 3: The difference between the multiple of the divisor and the first digit, that is 2 - 2 = 0
Step 4: Now we will bring down the next digit (5), forming 05. Divide 5 ÷ 2 = 2 remainder 1 (since 2 × 2 = 4, 5 – 4 = 1). The quotient digit is 2.
Step 5: Follow the steps till the remainder is 0 or less than the divisor.
So, Continuing, bring down the next digit (2), forming 12. Divide 12 ÷ 2 = 6, 12 – 12 = 0. Thus, 252 ÷ 2 = 126, remainder 0.
Case 2: When the first digit of the dividend is less than the divisor
In this section, let’s learn how to divide a number when the first digit of the dividend is less than the divisor. For example, dividing 235 by 5.
Step 1: If the first digit of the dividend is less than the divisor, that is when dividing 235 by 5, where 2 is less than the divisor 5. We cannot divide it, so we need to divide the first two digits, that is 23 by 5.
Step 2: As 5 × 4 = 20, the quotient is 4, and the leftover is 23 - 20 = 3.
Step 3: Bringing down the next digit which is 5, next to 3, so the new dividend is 35
Step 4: Repeat the steps till the remainder is 0 or less than the divisor.
Here, the quotient is 47, so 235 ÷ 5 = 47
Long Division without Remainder
In a long division without remainder, the remainder will always be zero. For long division without remainder, we follow the same steps. Dividing 354 ÷ 3
Step 1: Check whether the first two digits are greater than or equal to the divisor.
Step 2: Here, the quotient is 1, as 3 × 1 = 3. Subtracting 3 - 3 = 0.
Step 3: Bringing down the next number of the dividend, here it is 5.
5 ÷ 3 = 1 and the remainder is 2.
Step 4: Repeat the steps till the remainder is 0
Here, the quotient is 118 and the remainder is 0, so 354 ÷ 3 = 118.
Long Division by 2-Digit Number
We have learned how to divide a number when the divisor has two digits. The process is the same as dividing a number with a digit divisor but here we consider the first two digits of the dividend.
Step 1: Identify the first two digits of the dividend from the left-hand side. Check whether the digits are greater than or equal to the dividend.
For example, divide 363 by 11. Here 33 is greater than 11.
Step 2: Then find the multiple of the divisor which is equal to or smaller than the dividend.
As 11 × 3 = 33, so the quotient is 3.
Step 3: Calculating the difference between the multiple of the divisor and the dividend, we bring down the next digit of the dividend next to the difference.
Here the difference is 3, as 36 - 33 = 3.
Step 4: Continue the process till the remainder is 0 or less than the divisor.
363 ÷ 11 = 33.
Long Division of Polynomials
The expressions including the constants, variables, and exponents are polynomials. Now we can learn how to divide polynomials step by step.
Step 1: Arranging the terms in order, that is, both the dividend and the divisors are arranged in descending order of the exponents. That is the variables with higher exponents are arranged first and then the variables with lower exponents. Replace the missing term with 0. For example, let’s divide 3x2 + 4x3 + x - 4 by x - 2. It can be arranged as 4x3 + 3x2 + x - 4
Step 2: Dividing the first term of the dividend by the first term of the divisor, and the result is the first term in the quotient. That is 4x3/x = 4x2, so, 4x2 is the first term in quotient is 4x2
Step 3: Find the product of the divisor and first term in the quotient. That is 4x2 × (x - 2) = 4x3 - 8x2
Step 4: Find the difference between the result in step 3 and the dividend, the new value will be the new dividend. Here, it is (4x3 + 3x2 + x - 4) - (4x3 - 8x2) = 11x2 + x - 4.
Step 5: To find the new quotient repeat step 2 and step 4. The process is continued till we get the remainder which is zero or less than the divisor.
That is 11x2 / x = 11x, it is the next quotient.
Multiplying the new quotient with the divisor 11x (x -2) = 11x2 - 22x
So, the new dividend is (11x2 + x - 4) - (11x2 - 22x) = 23x - 4
The next quotient is 23x / x = 23
Multiplying the quotient with the divisor, 23 (x -2) = 23x - 46
That is (23x - 4) - (23x - 46) = -4 + 46 = 42
So, 3x2 + 4x3 + x - 4 by x - 2 = 4x2 + 11x + 23 + (42/x-2)
Long Division with Decimal
Decimals are a way of representing numbers: a whole number and a fraction, such as 252.12. Now let’s learn how to divide a decimal with a whole number.
For example, 42.8 ÷ 4
Step 1: Divide the first digit of the dividend with the divisor, that is 4 ÷ 4 = 1, so the quotient is 1. Subtract: 4 - 4 = 0
Step 2: Bring down the next digit from the dividend, that is 2 as 2 is smaller than 4 we cannot divide, so we write 0 in the quotient and bring down the next digit.
Step 3: Adding a decimal point in the quotient, as the next digit is a decimal value. So, the next dividend is 28
Step 4: Repeat the process till the remainder is 0 or less than the divisor.
Divide 28 ÷ 4 = 7, 28 – 28 = 0. The quotient is 10.7, and the remainder is 0. Therefore, 42.8 ÷ 4 = 10.7
Real-world applications of Long Division
In real life, we use the long division method from everyday problems to complex scientific calculations. Let’s discuss some real-world applications of the long division method.
- The long division method is used to split the bill among the group of people.
- For calculating the monthly payments of loans, we use the long division in banking.
- In computer science, the long division is used for data compression, digital commutation, and computer graphics.
- In calculus and algebra, the long division method is used to divide polynomials, find the GCF of polynomials, and factorial of polynomial.
Common Mistakes and How to Avoid Them in Long Division
Students usually make mistakes when working on the long division method as it is confusing. Mostly they repeat the same mistakes so let’s learn some common mistakes and the ways to avoid them in the long division method.
Mistake 1
Using the incorrect division order
Students sometimes start from the right-hand side, which can lead to error. So it is important to remember that the division always starts from the left-hand side. In the long division, it is important to mark the steps.
Mistake 2
Skipping steps in long division method
Skipping the steps in the long division method is another common mistake that students usually make. So when doing the long division, always write down each step, including division, multiplication, and subtraction, and make sure it's well aligned.
Mistake 3
Making errors in subtractions
When doing the subtraction in the long division method, especially when borrowing the digits. So always double-check the answer by doing the subtraction.
Mistake 4
Errors in the long division of polynomials
Long division of polynomials is kind of similar to the long division method, but mistakes happen by ignoring to align like terms and confusing with the missing degree. To avoid these students should remember that the dividend and divisor should be arranged in descending order and make sure to add a 0 coefficient in the missing degree.
Mistake 5
Not verifying the answer
When doing a long division method, errors are common among students as it involves various steps and operations. So once done with long division, always verify the answer, by multiplying the quotient by the divisor and the product should be the dividend.
Solved Examples of Long Division
Problem 1
A school has 864 students, and they are divided into 6 equal groups. How many students are in each group?
The number of students in each group is 144
Explanation
To find the number of students in each group we divide 864 by 6
864 ÷ 6 = 144
So, the number of students in each group is 144
Problem 2
Divide x^3 + 3x^2 − 4x − 12 by x + 2
x3 + 3x2 − 4x − 12 by x + 2 = x2 + x - 6
Explanation
To find the value x3 + 3x2 − 4x − 12 by x + 2 we use long division for polynomial
Dividing the first term of the dividend with the first term of that is x3 ÷ x = x2, it is the first term in the quotient.
Multiply the divisor with the first term quotient, that is x2 (x + 2) = x3 + 2x2
Subtracting the product from the dividend, (x3 + 3x2 - 4x - 12) - (x3 + 2x2) = x2 - 4x - 12
To find the new quotient, we divide x2 by x, that is x2 ÷ x = x
Multiplying the new quotient with the divisor, x(x + 2) = x2 + 2x
Subtracting the product from the dividend, (x2 - 4x - 12) - (x2 + 2x) = -6x - 12
The new quotient is -6x ÷ x = -6
Multiplying the new quotient with the divisor, -6(x + 2) = -6x -12
Subtracting the product from the dividend, (-6x - 12) - (-6x - 12) = 0
So, x3 + 3x2 − 4x − 12 by x + 2 = x2 + x - 6
Problem 3
A bakery made 126.4 kg of dough and divided it equally into 16 batches. How much dough is in each batch?
The amount of dough required for each batch is 7.9 kg
Explanation
To find the amount of dough needed we divide 126.4 by 16
That is 126.4 ÷ 16 = 7.9
The amount of dough required for each batch is 7.9 kg
Problem 4
A bookstore has 1,764 books arranged equally on 12 shelves. How many books are on each shelf?
The number of books on each shelf is 147
Explanation
To find the number of books on each shelf we divide 1764 by 12
That is 1764 ÷ 12 = 147
Problem 5
A company has 7,650 flyers to distribute equally among 25 locations. How many flyers go to each location?
The number of flyers for each location is 306
Explanation
To find the number of flyers distributed in each location = 7650 ÷ 25 = 306
FAQs on Long Division
1.What is a long division?
2.What are the 5 steps of long division?
3.What is 777 divided by 7?
4.What is the long division of polynomials?
5.Can we use the long-division method for decimals?
6.How can children in Philippines use numbers in everyday life to understand Long Division?
7.What are some fun ways kids in Philippines can practice Long Division with numbers?
8.What role do numbers and Long Division play in helping children in Philippines develop problem-solving skills?
9.How can families in Philippines create number-rich environments to improve Long Division skills?
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Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
