Long Division Calculator

Enter your dividend and divisor above to divide any two numbers instantly. The calculator shows the full long division problem worked out step by step, including the quotient, remainder, decimal, and fraction forms.

Long division can feel tedious by hand. This tool handles the repetitive divide, multiply, subtract cycle for you, one step at a time. You get a clear breakdown you can follow, check against your own work, or use to learn the method.

How the Long Division Calculator Works Step by Step

The calculator mirrors exactly what you would write on paper. It takes two numbers (the dividend and divisor), then walks through each digit of the dividend using the standard long division algorithm. Every intermediate step is visible so you can see how the final answer forms.

Divide, Multiply, Subtract, Repeat

Long division follows a four-part loop:

Divide the current portion of the dividend by the divisor. Write the result above the line.
Multiply that result by the divisor.
Subtract the product from the current portion.
Bring down the next digit from the dividend, then repeat.

This cycle continues until there are no more digits to bring down. If there is a leftover amount at the end, that is the remainder. The calculator labels each round of this loop so you can follow along neatly.

Dividend, Divisor, Quotient, and Remainder

A few terms appear throughout every long division problem:

Dividend: the number being divided (the larger number inside the division bracket).
Divisor: the number you are dividing by (outside the bracket).
Quotient: the answer, or the number of times the divisor fits into the dividend.
Remainder: the leftover part after the divisor no longer fits evenly.

For example, dividing 47 by 5 gives a quotient of 9 and a remainder of 2, because 5 × 9 = 45 with 2 left over.

Step-by-Step Long Division With Remainders

When a divisor does not go into the dividend evenly, you get a quotient with a remainder. The calculator shows this as a whole number plus the leftover value. From there, you can express the remainder as a fraction or a decimal, depending on what you need.

Long Division With Remainders as Fractions

To convert a remainder into a fraction, place the remainder over the divisor. The result is a mixed number.

Using the example above, 47 ÷ 5 = 9 remainder 2. As a fraction, that is 9 2/5. No extra calculation is needed. The calculator displays this form automatically.

Fractions are useful in math worksheets and problems that call for exact answers. They avoid rounding entirely.

Long Division With Remainders as Decimals

To express the remainder as a decimal, continue the long division process. Add a decimal point to the quotient and append a zero to the remainder, then keep dividing.

For 47 ÷ 5:

After reaching remainder 2, place a decimal point and bring down a 0, making it 20.
5 goes into 20 exactly 4 times.
The final answer is 9.4.

Some division problems produce decimals that terminate quickly. Others produce repeating decimals (covered below). The calculator handles both and labels the decimal result clearly.

Visual Long Division on a Number Line

A number line can make division more concrete, especially for learners. Imagine marking 0 at the left and the dividend at the right. Each jump along the line represents one group of the divisor's size.

For 47 ÷ 5, you would count jumps of 5: 0, 5, 10, 15 … 40, 45. That is 9 full jumps. The gap between 45 and 47 is the remainder of 2.

This visual approach reinforces what the quotient means: the number of groups that fit, with the remainder showing how far you are from the next full group. It is particularly helpful for building number sense alongside the written algorithm.

Long Division Problems: Digit by Digit

The core skill in long division is processing one digit at a time. The difficulty scales with the number of digits in the divisor.

Solving a Long Division Problem With a Single-Digit Divisor

Single-digit divisors (2 through 9) are where most students start. The key is knowing basic multiplication facts so each "divide" step is fast.

Example: 738 ÷ 6

6 into 7 → 1. Write 1 above the 7. Multiply: 1 × 6 = 6. Subtract: 7 − 6 = 1.
Bring down 3, making 13. 6 into 13 → 2. Multiply: 2 × 6 = 12. Subtract: 13 − 12 = 1.
Bring down 8, making 18. 6 into 18 → 3. Multiply: 3 × 6 = 18. Subtract: 18 − 18 = 0.

Quotient: 123 with no remainder.

Multi-Digit Division Problems

When the divisor has two or more digits, the "divide" step requires estimation. You ask: how many times does the divisor fit into the current group of digits?

Example: 4,536 ÷ 24

24 into 4 → 0. 24 into 45 → 1. Multiply: 1 × 24 = 24. Subtract: 45 − 24 = 21.
Bring down 3, making 213. 24 into 213 → 8. Multiply: 8 × 24 = 192. Subtract: 213 − 192 = 21.
Bring down 6, making 216. 24 into 216 → 9. Multiply: 9 × 24 = 216. Subtract: 216 − 216 = 0.

Quotient: 189 with no remainder.

The calculator removes the guesswork from estimation and shows each subtraction step neatly.

Decimal Long Division Step-by-Step

Decimal long division works the same way as whole-number division. The only extra detail is placing the decimal point correctly.

If the dividend has a decimal (for example, 9.84 ÷ 4), the calculator aligns the decimal point in the quotient directly above its position in the dividend, then proceeds digit by digit.

If the divisor has a decimal, the calculator first shifts both numbers so the divisor becomes a whole number. For example, 7.2 ÷ 0.6 becomes 72 ÷ 6. This rewrite does not change the answer; it just makes the steps easier to follow.

Repeating Decimals in Long Division

Some division problems never terminate. When you see the same remainder appear a second time, the decimal digits will repeat in a cycle.

Example: 10 ÷ 3

3 into 10 → 3, remainder 1. Bring down 0 → 10 again.
The remainder of 1 keeps recurring, so the answer is 3.333… or 3.3̄.

A decimal repeats because the set of possible remainders is finite. If the divisor is d, only remainders 0 through d − 1 exist. Once a remainder reappears, the cycle locks in. The calculator identifies the repeating block and marks it for you.

Polynomial Long Division Calculator

This calculator also supports polynomial long division, where you divide one polynomial expression by another.

Enter expressions like (x³ + 2x² − 5x + 3) ÷ (x − 1). The calculator returns the quotient polynomial and any remainder, laid out step by step.

How the Polynomial Long Division Calculator Differs From Number Division

The algorithm is the same (divide, multiply, subtract, bring down the next term), but instead of digits, you work with terms ordered by descending exponent.

Key differences:

Divide the leading term of the current polynomial by the leading term of the divisor.
Multiply the entire divisor by that result.
Subtract term by term, combining like terms.
Repeat until the degree of the remaining polynomial is less than the degree of the divisor.

Missing powers (for example, no x term) must be represented with a zero coefficient. The calculator inserts these automatically so the alignment stays correct.

Polynomial long division is essential for factoring higher-degree expressions, finding oblique asymptotes, and simplifying rational equations.

Math Behind Long Division

Long division is not a separate operation from basic arithmetic. It is a structured way of combining multiplication and subtraction to solve a division problem one piece at a time.

Multiplication and Subtraction in Every Step

Each cycle tests a guess: "How many times does the divisor fit here?" You verify the guess by multiplying, then subtract to find what is left. If the subtraction result is negative, the guess was too high. If the result is larger than the divisor, the guess was too low.

This self-correcting loop is why the algorithm works reliably for any size of numbers. It reduces a large division problem into a series of smaller, manageable calculations.

Euclidean Division and Integer Quotients

Formally, long division implements the Euclidean division theorem. For any integers a (dividend) and b (divisor, not zero), there exist unique integers q (quotient) and r (remainder) such that:

a = b × q + r, where 0 ≤ r < |b|.

This guarantees that every whole-number division has exactly one correct quotient and remainder pair. It is also the foundation of the Euclidean algorithm for finding the greatest common factor of two numbers.