Long Division

Long Division is a method for dividing large numbers, which breaks the division problem into multiple steps following a sequence. Just like the regular division problems, the dividend is divided by the divisor which gives a result known as the quotient, and sometimes it gives a remainder too. This article will give you an overview of the long division method along with its steps and examples.

In math, long division is a method for dividing large numbers into steps or parts, breaking the division problem into a sequence of easier steps. It is the most common method used to solve problems based on division. Observe the following long division to see the divisor, the dividend, the quotient, and the remainder.

As you have seen above, while performing the steps of long division, there is an equation formed which is known as the long division equation. For example, while dividing 75 by 4, we get 75 = 4 × 18 + 3 where 75 is the dividend, 4 is the divisor, 18 is the quotient, and 3 is the remainder. The general form of a long division equation is "Dividend = Divisor × Quotient + Remainder". Here are the terms related to a division which are also considered as the parts of long division. They are the same terms that are used in the regular division.

• Dividend
• Divisor
• Quotient
• Remainder

Have a look at the table given below in order to understand the terms related to the long division with reference to the example shown above.

The division is one of the four basic mathematical operations, the other three being addition, subtraction, and multiplication. In arithmetic, long division is a standard division algorithm for dividing large numbers, breaking down a division problem into a series of easier steps.

Long Division Steps

To perform division requires the construction of a tableau. The divisor is separated from the dividend by a right parenthesis 〈)〉 or vertical bar 〈|〉 and the dividend is separated from the quotient by a vinculum (an overbar). Now, let us follow the steps of the long division given below to understand the process.

• Step 1: Take the first digit of the dividend from the left. Check if this digit is greater than or equal to the divisor.
• Step 2: Then divide it by the divisor and write the answer on top as the quotient.
• Step 3: Subtract the result from the digit and write the difference below.
• Step 4: Bring down the next digit of the dividend (if present).
• Step 5: Repeat the same process.

Let's have a look at the examples given below for a better understanding of the concept.

Case 1: When the first digit of the dividend is equal to or greater than the divisor.

Let's consider an example: Divide 435 ÷ 4. The steps of long division are given below:

• Here, the first digit of the dividend is 4 and it is equal to the divisor. So, 4 ÷ 4 = 1. So, 1 is written on top as the first digit of the quotient.
• Subtract: 4 - 4 = 0.
• Bring the second digit of the dividend down and place it besides 0.
• Now, 3<4. Hence, we write 0 as the quotient and bring down the next digit of the dividend and place it besides 3.
• Now, we have 35 as the new dividend. 35 > 4. 35 is not divisible by 4, so we look for the number just less than 35 in the table of 4. We know that 4 × 8 = 32 < 35 so, we go for it.
• Write 8 in the quotient. Subtract: 35 - 32 = 3.
• 3<4. Thus, 3 is the remainder and 108 is the quotient.

Case 2: When the first digit of the dividend is less than the divisor.

Let's consider another example: Divide 735 ÷ 9.

• Since the first digit of the dividend is less than the divisor, put zero as the quotient and bring down the next digit of the dividend. Now consider the first 2 digits to proceed with the division.
• 73 is not divisible by 9 but we know that 9 × 8 = 72 so, we go for it.
• Write 8 in the quotient and subtract 73 - 72 = 1.
• Bring down 5. The number to be considered now is 15.
• Since 15 is not divisible by 9 but we know that 9 × 1 = 9, so, we take 9.
• Subtract: 15 - 9 = 6. Write 1 in the quotient.
• Now, 6<9. Thus, remainder = 6 and quotient = 81.

Case 3: When the divisor doesn't go with the first digit of the dividend.

Let's solve one more example: Divide 3640 ÷ 15.

• Since the first digit of the dividend is not divisible by the divisor, we consider the first two digits (36).
• Now, 36 is not divisible by 15 but 15 × 2 = 30, so, write 2 as the first digit in the quotient.
• Write 30 below 36 and subtract 36 - 30 = 6.
• Since 6<15, we will bring down 4 from the dividend to make it 64.
• 64 is not divisible by 15 but 15 × 4 = 60, so, write 4 in the quotient.
• Write 60 below 64 and subtract 64 - 60 = 4.
• Since 4<15, bring down 0 from the dividend to make it 40.
• Since 40 is not divisible by 15 but 15 × 2 = 30, so, write 2 in the quotient.
• Write 30 below 40 and subtract 40 - 30 = 10.
• Now 10<15. Thus, remainder = 10 and quotient = 242.

Long division problems also include problems related to long division polynomials and long division with decimals.

Long Division of Polynomials

When there are no common factors between the numerator and the denominator, or if you can't find the factors, you can use the long division process to simplify the expression. For more details about long division polynomials, visit the Dividing Polynomials page.

Long Division with Decimals

Long division with decimals can be easily done just like the normal division. For more details about long division with decimals, visit the Dividing Decimals page.

Given below are a few important tips and tricks that would help you while working with long division:

• In the case of whole numbers, the dividend is always greater than or equal to the divisor and the quotient.
• The remainder is always smaller than the divisor.
• For division, the divisor cannot be 0.
• The division is repeated subtraction, so we can check our quotient by repeated subtractions as well.
• We can verify the quotient and the remainder of the division using the division formula: Dividend = (Divisor × Quotient) + Remainder.
• If the remainder is 0, then we can check our quotient by multiplying it with the divisor. If the product is equal to the dividend, then the quotient is correct.

Related Articles

Check these interesting articles related to the long division method.

• Long Division Formula
• Long Division of Polynomials
• Long Division Without Remainders Worksheets
• Long Division Calculator

What is Long Division in Math?

Long division is a process to divide large numbers in a convenient way. The number which we divide into smaller groups is known as a dividend, the number by which we divide it is called the divisor, the value got after doing the division is the quotient, and the number left after the division is called the remainder.

How to do Long Division?

The following steps explain the process of long division:

• Write the dividend and the divisor at their respective positions.
• Take the first digit of the dividend from the left.
• If this digit is greater than or equal to the divisor, then divide it by the divisor and write the answer on top as the quotient.
• Write the product below the dividend and subtract the result from the dividend to get the difference. If this difference is less than the divisor, and there are no numbers left in the dividend, then this is considered to be the remainder and the division is done. However, if there are more digits in the dividend to be carried down, we continue with the same process until there are no more digits left in the dividend.

What are the Steps of Long Division?

Given below are the 5 main steps of long division. For example, let us see how we divide 52 by 2.

• Step 1: Consider the first digit of the dividend which is 5 in this example. Here, 5 > 2. 5 is not divisible by 2.
• Step 2: We know that 2 × 2 = 4, so, we write 2 as the quotient.
• Step 3: 5 - 4 = 1 and 1 < 2 (After writing the product 4 below the dividend, we subtract them).
• Step 4: 1 < 2, so we bring down 2 from the dividend and we get 12 as the new dividend now.
• Step 5: Repeat the process till the time you get a remainder less than the divisor. 12 is divisible by 2 as 2 × 6 = 12, so we write 6 in the quotient, and 12 - 12 = 0 (remainder).

Therefore, the quotient is 26 and the remainder is 0.

How do you do Long Division with 2 Digits?

Consider both the digits of the divisor and check for the divisibility of the first two digits of the dividend. If the first 2 digits of the dividend are less than the divisor, then consider the first three digits of the dividend. Proceed with the division in the same way as we divide regular numbers.

What is the Long Division of Polynomials?

In algebra, the long division of polynomials is an algorithm to divide a polynomial by another polynomial of the same or the lower degree. For example, (4x² - 5x - 21) is a polynomial that can be divided by (x - 3) following some defined rules, which will result in 4x + 7 as the quotient.

How to do Long Division with Decimals?

The long division with decimals is performed in the same way as the normal division. It follows the steps given below:

• Write the division in the standard form.
• Start by dividing the whole number part by the divisor.
• Place the decimal point in the quotient above the decimal point of the dividend.
• Bring down the tens place digit.
• Divide and bring down the other digit in sequence.
• Divide until all the digits of the dividend are over and a number less than the divisor or 0 is obtained in the remainder.

What is Long Division Symbol Called?

The divisor and the dividend are separated by a right parenthesis 〈)〉 or a vertical bar 〈|〉 whereas, the dividend and the quotient are separated by a vinculum or an overbar. The combination of these two symbols is referred to as long division symbol or division bracket.

What is the Remainder in Polynomial Long Division?

While dividing polynomials, sometimes there is a value left that is lesser in degree as compared to the divisor, that value is known as remainder. It can be in the form of an expression or a number.