Long Division Calculator — With Steps
Enter any two numbers and instantly see the quotient, remainder, decimal result, and a full step-by-step solution. Free, fast, and built by He Loves Math.
Divide Two Numbers
What Is Long Division?
Long division is a standard arithmetic algorithm for dividing large numbers that cannot be solved mentally. It breaks a complex division problem into a series of simpler steps — divide, multiply, subtract, bring down — and repeats the cycle until every digit of the dividend has been processed. The method is taught in elementary school (typically grades 3–5) and remains essential for algebra, polynomial division, and even computer-science algorithms.
In its most basic notation, we write a division problem as:
where \(a\) is the dividend (the number being divided), \(b\) is the divisor (the number you divide by), \(q\) is the quotient (the whole-number result), and \(r\) is the remainder (what is left over). Equivalently:
This identity — known as the Division Algorithm — is one of the cornerstones of number theory and is formalised in the Euclidean division theorem.
How to Do Long Division — Step by Step
Follow these steps to divide any integer by another using the long-division method. We will use the example \(7853 \div 12\).
- Set up the problem. Write the divisor (\(12\)) to the left of a division bracket and the dividend (\(7853\)) underneath the bracket.
- Divide. Look at the first digit(s) of the dividend that the divisor can fit into. \(12\) does not fit into \(7\), so take the first two digits: \(78\). Ask: how many times does \(12\) go into \(78\)? The answer is \(6\) because \(12 \times 6 = 72\). Write \(6\) above the bracket.
- Multiply. Compute \(12 \times 6 = 72\) and write it below \(78\).
- Subtract. \(78 - 72 = 6\). Write \(6\) below the line.
- Bring down the next digit (\(5\)) to get \(65\).
- Repeat. \(12\) goes into \(65\) five times (\(12 \times 5 = 60\)). Write \(5\) in the quotient. Subtract: \(65 - 60 = 5\). Bring down the next digit (\(3\)) to get \(53\).
- Continue. \(12\) goes into \(53\) four times (\(12 \times 4 = 48\)). Subtract: \(53 - 48 = 5\). No more digits to bring down, so the remainder is \(5\).
- Write the answer. The quotient is \(654\) with a remainder of \(5\), i.e. \(7853 \div 12 = 654 \;\text{R}\; 5\).
You can verify: \(12 \times 654 + 5 = 7848 + 5 = 7853\). ✓
The Division Algorithm — Mathematical Foundation
Every long-division problem is an application of the Division Algorithm, which states:
The uniqueness of \(q\) and \(r\) guarantees that every division problem has exactly one correct quotient-remainder pair. This theorem underpins modular arithmetic, the Euclidean algorithm for finding the GCD, and much of modern cryptography.
Converting the Remainder to a Decimal
If you want a decimal answer instead of a remainder, continue the long-division process by appending a decimal point and zeros to the dividend:
For our example: \(\frac{7853}{12} = 654 + \frac{5}{12} = 654.41\overline{6}\). The bar over the \(6\) indicates a repeating decimal.
Worked Examples
Example 1 — Simple Division with No Remainder
Compute \(1296 \div 6\).
Verification: \(6 \times 216 = 1296\). ✓ Since the remainder is zero, we say \(6\) divides \(1296\) evenly, or \(6 \mid 1296\).
Example 2 — Division with a Remainder
Compute \(5471 \div 13\).
Verification: \(13 \times 420 + 11 = 5460 + 11 = 5471\). ✓
Example 3 — Dividend Smaller Than Divisor
Compute \(7 \div 15\).
When the dividend is smaller than the divisor, the quotient is always \(0\) and the remainder equals the dividend itself.
Example 4 — Large Numbers
Compute \(123456 \div 789\).
Verification: \(789 \times 156 + 372 = 123084 + 372 = 123456\). ✓
When and Why Long Division Matters
- Polynomial division: In algebra and calculus, long division of polynomials follows the exact same divide-multiply-subtract-bring-down pattern.
- Converting fractions to decimals: Long division is the standard method for converting any fraction \(\frac{a}{b}\) into its decimal representation.
- Understanding remainders and modular arithmetic: Modern cryptography (RSA, Diffie-Hellman) relies on modular arithmetic, which is simply the remainder part of division.
- Mental math scaffolding: Fluency with long division strengthens number sense, estimation, and the ability to check calculator results for reasonableness.
- Standardised tests: The SAT, ACT, GRE, and many state assessments include division-based problems where showing your work is essential.
Common Mistakes to Avoid
- Forgetting to place a 0 in the quotient when the divisor does not fit into the current number after bringing down.
- Subtraction errors — a single wrong subtraction cascades into an incorrect final answer.
- Not bringing down all digits — skipping a digit changes the problem entirely.
- Dividing by zero — division by zero is undefined in mathematics. Our calculator catches this and shows an error.
- Ignoring negative signs — when one (but not both) of the numbers is negative, the quotient is negative.
Quick Reference Table — Common Divisions
| Dividend | Divisor | Quotient | Remainder | Decimal |
|---|---|---|---|---|
| 100 | 3 | 33 | 1 | 33.333… |
| 250 | 7 | 35 | 5 | 35.714… |
| 1000 | 6 | 166 | 4 | 166.667… |
| 999 | 9 | 111 | 0 | 111 |
| 5000 | 13 | 384 | 8 | 384.615… |
| 7777 | 11 | 707 | 0 | 707 |
| 12345 | 67 | 184 | 17 | 184.254… |
| 100000 | 37 | 2702 | 26 | 2702.703… |
Long Division vs. Short Division
Short division is a condensed form of long division used when the divisor is a single digit (1–9). Instead of writing every multiplication and subtraction step, you carry remainders mentally. For example, \(846 \div 3\): 8÷3=2 r2 → 24÷3=8 → 6÷3=2 → answer 282. Short division is faster but only practical for single-digit divisors. For multi-digit divisors, long division is essential.
Relationship Between Division and Other Operations
Division is the inverse of multiplication. Understanding this relationship helps you check your work:
Division is also connected to fractions and ratios:
In more advanced mathematics, division extends to rational numbers, real numbers, and even complex numbers, always following the same fundamental principle.
Frequently Asked Questions (FAQs)
Long division is a step-by-step method for dividing large numbers. You repeatedly divide, multiply, subtract, and bring down the next digit until all digits have been processed. The result is a quotient and a remainder.
Multiply the quotient by the divisor and add the remainder. If the result equals the original dividend, your answer is correct: \(b \times q + r = a\).
No. Division by zero is undefined in mathematics. There is no number that, when multiplied by zero, gives a non-zero result. Our calculator will display an error if you attempt to divide by zero.
A remainder is the whole-number leftover after division. A decimal is that remainder expressed as a fraction of the divisor. For instance, \(7 \div 2 = 3 \;\text{R}\; 1\), or \(3.5\) in decimal, because the remainder \(1\) divided by the divisor \(2\) is \(0.5\).
The four parts are: the dividend (number being divided), the divisor (number you divide by), the quotient (result), and the remainder (leftover). In \(17 \div 5 = 3 \;\text{R}\; 2\), 17 is the dividend, 5 is the divisor, 3 is the quotient, and 2 is the remainder.
Divide the absolute values using the standard algorithm. Then apply the sign rule: if exactly one of the dividend or divisor is negative, the quotient is negative. If both are negative or both positive, the quotient is positive.
The Division Algorithm is a theorem stating that for any integer \(a\) and positive integer \(b\), there exist unique integers \(q\) and \(r\) such that \(a = bq + r\) and \(0 \le r < b\). It is the mathematical foundation of long division.
This calculator works with whole numbers (integers). If you enter decimal inputs, they will be rounded to the nearest integer before the division is performed. For fraction-to-decimal conversions, use our dedicated fraction calculator.
Long division is used for splitting costs, converting measurements, calculating averages, understanding interest rates, dividing recipes, and many engineering and science applications. It also forms the foundation for polynomial long division in algebra.
For single-digit divisors, short division is a faster method where you carry remainders mentally. For multi-digit divisors, estimation and rounding can speed up each "divide" step, but the full algorithm must still be followed for accuracy.
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