What Is Long Division?
Long division is a method for dividing large numbers that breaks the problem into a series of smaller, more manageable steps. Unlike short division (which you can do in your head for simple problems), long division provides a structured written process that works for any division problem, no matter how large the numbers.
The four key terms in division are: the dividend (the number being divided), the divisor (the number you are dividing by), the quotient (the answer), and the remainder (what is left over if the division is not exact). In the problem 845 / 7, 845 is the dividend, 7 is the divisor, the quotient is 120, and the remainder is 5.
Long division follows a repeating cycle of four steps: Divide, Multiply, Subtract, Bring down. Some students remember this with the mnemonic DMSB — "Dad, Mom, Sister, Brother" or "Does McDonald's Sell Burgers." Once you internalize this cycle, you can divide any number by any other number.
The Four Steps: Divide, Multiply, Subtract, Bring Down
Let's walk through 936 / 4 to demonstrate each step. Set up the problem with the division bracket: 4 goes into 936.
Step 1 — Divide: Look at the first digit of the dividend (9). How many times does 4 go into 9? It goes in 2 times (because 4 x 2 = 8, and 4 x 3 = 12 which is too big). Write 2 above the 9 in the quotient position.
Step 2 — Multiply: Multiply the divisor (4) by the number you just wrote (2). 4 x 2 = 8. Write 8 below the 9.
Step 3 — Subtract: Subtract 8 from 9. 9 - 8 = 1. Write 1 below the line.
Step 4 — Bring down: Bring down the next digit of the dividend (3) to sit next to the 1, making 13. Now repeat the cycle: 4 goes into 13 three times (4 x 3 = 12). Write 3 in the quotient. Multiply: 4 x 3 = 12. Subtract: 13 - 12 = 1. Bring down the next digit (6), making 16. Divide: 4 goes into 16 exactly 4 times. Multiply: 4 x 4 = 16. Subtract: 16 - 16 = 0. No more digits to bring down, and the remainder is 0. The answer is 234.
Long Division with Remainders
Not every division comes out evenly. When you have completed all the steps and there is still a number left after the final subtraction, that number is the remainder. Let's solve 529 / 3.
3 goes into 5 one time (3 x 1 = 3). Subtract: 5 - 3 = 2. Bring down the 2, making 22. 3 goes into 22 seven times (3 x 7 = 21). Subtract: 22 - 21 = 1. Bring down the 9, making 19. 3 goes into 19 six times (3 x 6 = 18). Subtract: 19 - 18 = 1. No more digits to bring down. The answer is 176 remainder 1, or 176 R1.
You can express remainders in three ways: as a whole number remainder (176 R1), as a fraction (176 1/3, where the remainder goes over the divisor), or as a decimal (176.333..., by continuing the division with decimal places). Which format you use depends on the context of the problem.
To check your answer, use the inverse operation: multiply the quotient by the divisor and add the remainder. 176 x 3 = 528, plus remainder 1 = 529. This matches the original dividend, confirming the answer is correct. Always check your long division this way on tests.
When the Divisor Is Larger Than a Digit
When the divisor has two or more digits, the process is the same — but the "Divide" step requires estimation. Let's solve 4,752 / 23.
23 does not go into 4 (too small), so look at the first two digits: 47. How many times does 23 go into 47? Estimate: 23 x 2 = 46 (fits). 23 x 3 = 69 (too big). So 23 goes into 47 twice. Write 2 above the 7. Multiply: 23 x 2 = 46. Subtract: 47 - 46 = 1. Bring down the 5, making 15.
23 goes into 15 zero times. Write 0 in the quotient. Multiply: 23 x 0 = 0. Subtract: 15 - 0 = 15. Bring down the 2, making 152. How many times does 23 go into 152? Estimate: 23 x 6 = 138, 23 x 7 = 161 (too big). So 23 goes in 6 times. Multiply: 23 x 6 = 138. Subtract: 152 - 138 = 14. No more digits to bring down. The answer is 206 R14, or 206 14/23.
The estimation step is where most students struggle with multi-digit divisors. A helpful technique is to round the divisor and use that to estimate. For 23, round to 20. 20 goes into 152 about 7 times — but check 23 x 7 = 161, which is too big, so go with 6. With practice, this estimation becomes fast and intuitive.
Decimal Division: Continuing Past the Remainder
Instead of stopping at a remainder, you can continue dividing to get a decimal answer. The key insight is that you can add zeros after the decimal point in the dividend without changing its value (845 = 845.000...).
Let's solve 845 / 7 as a decimal. First, do the whole number division: 7 into 8 is 1 (remainder 1). Bring down 4, making 14. 7 into 14 is 2 (remainder 0). Bring down 5. 7 into 5 is 0 (remainder 5). Now, instead of writing R5, place a decimal point in the quotient (after 120) and add a zero to make the remainder 50.
7 into 50 is 7 (7 x 7 = 49). Subtract: 50 - 49 = 1. Add another zero: 10. 7 into 10 is 1 (7 x 1 = 7). Subtract: 10 - 7 = 3. Add another zero: 30. 7 into 30 is 4 (7 x 4 = 28). Subtract: 30 - 28 = 2. The decimal continues: 845 / 7 = 120.714285... (the digits 714285 repeat forever).
For most homework problems, you will be told how many decimal places to round to. If the problem says "round to two decimal places," calculate three decimal places and round the last one. For 845 / 7, rounded to two decimal places: 120.71.
Dividing Decimals by Whole Numbers
When the dividend is a decimal (like 18.6 / 4), the process is the same as whole number long division. The only extra step is placing the decimal point in the quotient directly above the decimal point in the dividend.
Solve 18.6 / 4: Set up the division bracket. 4 into 18 is 4 (4 x 4 = 16). Subtract: 18 - 16 = 2. Place the decimal point in the quotient above the one in the dividend. Bring down the 6, making 26. 4 into 26 is 6 (4 x 6 = 24). Subtract: 26 - 24 = 2. You can continue by adding a zero: 20. 4 into 20 is 5. Answer: 4.65.
When the divisor is a decimal (like 18.6 / 0.4), convert the problem so the divisor is a whole number. Multiply both the dividend and divisor by 10 (or 100, or 1000 — whatever makes the divisor a whole number). 18.6 / 0.4 becomes 186 / 4 = 46.5. This works because multiplying both numbers by the same value does not change the quotient.
Common Mistakes in Long Division
Forgetting to write zero in the quotient is the number one long division mistake. When the divisor does not go into the current number (like when 23 does not go into 15 in our earlier example), you must write 0 in the quotient and bring down the next digit. Skipping this zero shifts all subsequent digits and gives a completely wrong answer.
Subtraction errors are the second most common problem. Long division involves many subtraction steps, and one error cascades through the rest of the problem. Double-check each subtraction, especially when borrowing is involved.
Misaligning digits causes problems too. Keep your columns straight — ones under ones, tens under tens. Using graph paper or turning lined paper sideways so the lines create columns can help enormously.
Finally, forgetting to bring down the next digit before dividing is a common procedural error. After subtracting, always bring down before you divide again. If you try to divide the remainder without bringing down, you will get the wrong quotient digit.
Practice Problems with Solutions
Problem 1: 672 / 8. Solution: 8 into 67 is 8 (64). Remainder 3, bring down 2, making 32. 8 into 32 is 4. Answer: 84.
Problem 2: 1,547 / 6. Solution: 6 into 15 is 2 (12). Remainder 3, bring down 4, making 34. 6 into 34 is 5 (30). Remainder 4, bring down 7, making 47. 6 into 47 is 7 (42). Remainder 5. Answer: 257 R5, or 257 5/6.
Problem 3: 3,024 / 12. Solution: 12 into 30 is 2 (24). Remainder 6, bring down 2, making 62. 12 into 62 is 5 (60). Remainder 2, bring down 4, making 24. 12 into 24 is 2. Answer: 252.
Problem 4: 95.4 / 6, rounded to two decimal places. Solution: 6 into 9 is 1 (6). Remainder 3, bring down 5, making 35. 6 into 35 is 5 (30). Remainder 5. Decimal point. Bring down 4, making 54. 6 into 54 is 9. Bring down 0, making 0. Answer: 15.90. If you need more practice or want the AI to walk you through a specific division problem, try ScanSolve — just snap a photo of the problem.