What Is Long Division?

Long division is a step-by-step method for dividing large numbers by hand. It breaks down the division problem into a series of simpler operations — divide, multiply, subtract, bring down — repeated until you've worked through every digit of the dividend. The result consists of a quotient (how many times the divisor fits into the dividend) and a remainder (what's left over).

Long division is expressed using the notation: dividend ÷ divisor = quotient remainder R, or equivalently: dividend = (quotient × divisor) + remainder.

For example: 256 ÷ 7 = 36 remainder 4, which means 256 = (36 × 7) + 4 = 252 + 4 = 256. ✓

The four key terms you'll use throughout every long division problem:

How to Do Long Division – Step-by-Step Guide

The long division algorithm follows a repeating four-step cycle: Divide → Multiply → Subtract → Bring Down. Here's a complete worked example: 845 ÷ 4

<h3>Step 1: Set Up</h3>
<p>Write the dividend (845) inside the division bracket and the divisor (4) outside to the left. You'll work from left to right through the digits of 845.</p>

<h3>Step 2: Divide the First Digit</h3>
<p>Look at the first digit: 8. Ask: how many times does 4 go into 8? Answer: 2. Write 2 above the 8.</p>

<h3>Step 3: Multiply</h3>
<p>Multiply 2 × 4 = 8. Write 8 below the 8 in the dividend.</p>

<h3>Step 4: Subtract</h3>
<p>8 − 8 = 0. Write 0 below.</p>

<h3>Step 5: Bring Down</h3>
<p>Bring down the next digit (4) to make 04. Ask: how many times does 4 go into 4? Answer: 1. Write 1 above the 4.</p>

<h3>Step 6: Repeat</h3>
<p>Multiply 1 × 4 = 4. Subtract: 4 − 4 = 0. Bring down the last digit (5). Ask: how many times does 4 go into 5? Answer: 1 (4 goes once). Write 1 above the 5. Multiply 1 × 4 = 4. Subtract: 5 − 4 = 1. No more digits to bring down.</p>

<h3>Result</h3>
<p><strong>845 ÷ 4 = 211 remainder 1</strong>. Check: 211 × 4 + 1 = 844 + 1 = 845. ✓</p>
<p>As a decimal: 845 ÷ 4 = 211.25 (add a decimal point and continue with zeros as needed).</p>

Long Division Examples – Worked Problems

Practice these examples to build fluency with long division. Each demonstrates a different aspect of the algorithm.

Notice that when the remainder is 0, the division is exact — the divisor divides the dividend evenly. When the remainder is nonzero, you can express the result as a mixed number (quotient + remainder/divisor) or a decimal.

Long Division with Decimals

To continue a long division problem past the decimal point, simply add a decimal point after the quotient and continue the algorithm by adding zeros to the right of the dividend's remainder.

Example: 22 ÷ 7

1. 22 ÷ 7 = 3 remainder 1 → write 3, bring decimal point
2. 10 ÷ 7 = 1 remainder 3 → write .1
3. 30 ÷ 7 = 4 remainder 2 → write .14
4. 20 ÷ 7 = 2 remainder 6 → write .142
5. 60 ÷ 7 = 8 remainder 4 → write .1428
6. 40 ÷ 7 = 5 remainder 5 → write .14285
7. 50 ÷ 7 = 7 remainder 1 → write .142857 (repeating!)

22 ÷ 7 = 3.142857142857… (the digits repeat with period 6). Notice this is very close to π ≈ 3.14159… (a coincidence!)

A fraction terminates in decimal form if and only if the denominator's only prime factors are 2 and/or 5. All other fractions produce repeating decimals.

Division Rules and Divisibility Tests

Before starting long division, check divisibility rules to quickly determine if division will be exact (remainder = 0). These rules save time and help catch errors.

Common Mistakes in Long Division

Long division errors typically fall into a few predictable patterns. Being aware of these helps you self-check and avoid them.

• Forgetting to bring down a digit: After each subtraction, bring down the very next digit before continuing. Skipping this step shifts all subsequent digits incorrectly.
• Writing 0 in the quotient: If the divisor doesn't go into the current partial dividend, write 0 in the quotient and bring down the next digit. Forgetting the 0 produces a quotient that's too small by a factor of 10.
• Estimation errors: If you estimate too high (e.g., 9 when 8 fits), the subtraction produces a negative number — back up and reduce your estimate by 1.
• Not checking your answer: Always verify: quotient × divisor + remainder = dividend. This 5-second check catches virtually every error.
• Misplacing the decimal point: When extending to decimals, bring the decimal point directly up to the quotient line and continue with zeros.

Long Division in Real Life

Long division skills translate directly to everyday quantitative tasks:

• Splitting costs: If 7 friends share a $256 restaurant bill, each pays $256 ÷ 7 = $36.57 (36 remainder 4, so someone pays $1 more).
• Unit conversion: Converting 1,000 miles to kilometers: 1,000 × 1.60934 = 1,609.34 km. Long division helps work through multi-digit multiplications and divisions without a calculator.
• Cooking and recipes: Scaling a recipe that serves 6 up to serve 22: divide each ingredient by 6, then multiply by 22 — or use long division to find the scaling factor 22 ÷ 6 = 3 remainder 4, i.e., multiply by 3⅔.
• Pace calculations: A runner covers 26.2 miles in 3 hours and 45 minutes (225 minutes). Average pace: 225 ÷ 26.2 = 8 minutes 35 seconds per mile (via long division: 225 ÷ 26 = 8 remainder 17; carry remainder to seconds: 17 × 60 / 26 ≈ 39 seconds).
• Estimating fuel stops: A car with a 45-liter tank gets 13 km/L. Range = 45 × 13 = 585 km. How many stops for a 2,000 km road trip? 2,000 ÷ 585 = 3 remainder 245, so 3 refueling stops plus a partial tank at the end.

Long Division vs Short Division vs Synthetic Division

Division Algorithms in Computing

The long division algorithm that students learn in school is directly implemented (in optimized form) in computer processors and programming languages. Understanding the algorithm illuminates how modern hardware works:

• Integer division in programming: In most languages, the / operator on integers performs truncating division (same as long division quotient), and the % operator returns the remainder (modulo). In Python: 256 // 7 = 36 and 256 % 7 = 4.
• Hardware division: CPUs implement division using restoring division, non-restoring division, or SRT (Sweeney Robertson Tocher) algorithms — all variations of long division done in binary. A typical integer division instruction takes 20–90 clock cycles on modern CPUs, making it one of the most expensive arithmetic operations.
• Floating-point division: Dividing decimal numbers in computers uses IEEE 754 floating-point arithmetic — a binary version of long division with a fixed number of significant bits (23 bits for float32, 52 bits for float64).
• Modular arithmetic: The remainder from long division (modulo operation) is foundational to hash functions, cryptographic algorithms, checksum validation (credit card Luhn algorithm), and calendar calculations.

Clock arithmetic example: what day of the week is 100 days from Tuesday (day 2, where Sunday=0)? (2 + 100) mod 7 = 102 mod 7. Long division: 102 ÷ 7 = 14 remainder 4. So day 4 = Thursday. This is the same long division algorithm — applied to circular (modular) arithmetic.

Division Table: 1 Through 12 Reference

This multiplication-division reference table covers 1–12 × 1–12. Use it to quickly verify long division quotients during manual calculations. Each cell shows a ÷ b (where a is the row heading multiplied by the column heading).

To use this table for long division: when dividing by 7, scan the "7" row to find the largest product that fits into your current partial dividend. If dividing into 47, look at the 7 row: 7×6=42, 7×7=49 — so 6 goes in with remainder 47−42=5. This is exactly the "estimate" step in long division. The multiplication table is the engine that powers long division.

Estimating Division Results Before Calculating

Strong estimation skills reduce the chance of major errors in long division. Before starting a problem, estimate the magnitude of the quotient using powers of 10 and rounding. This gives you a "sanity check" to catch mistakes.

• Round to the nearest power of 10: For 4,567 ÷ 23, round to 5,000 ÷ 25 = 200. This tells you the quotient should be in the hundreds — a two-to-three digit number. If your calculated quotient is 1,982 or 18, something went wrong.
• Use benchmark divisors: Dividing by 25 is the same as multiplying by 4 and dividing by 100. Dividing by 50 is multiplying by 2 and dividing by 100. Dividing by 125 is multiplying by 8 and dividing by 1000. Example: 750 ÷ 25 = (750 × 4) / 100 = 3000/100 = 30.
• Factor the divisor: If the divisor has convenient factors, divide in steps. 756 ÷ 28: divide by 4 first (756 ÷ 4 = 189), then by 7 (189 ÷ 7 = 27). This often requires smaller mental calculations than tackling 28 directly.
• Approximate first digit of quotient: The first digit of the quotient is always ⌊(first digit(s) of dividend) / divisor⌋. For 3,456 ÷ 8: first digit of 3,456 is 3; 3 ÷ 8 = 0, so look at 34 ÷ 8 = 4. Quotient starts with 4xx.

Practice estimation before calculating: it builds number sense and is the foundation of mental math. Professional mathematicians and engineers estimate before computing, not after — getting the order of magnitude right first prevents the most costly errors (being off by a factor of 10 or 100). A useful self-check: after completing long division, multiply your quotient by the divisor and add the remainder. If you get the original dividend, your calculation is correct. This multiplication serves as a fast verification and reinforces the inverse relationship between multiplication and division — understanding one operation deeply makes the other more intuitive.