What do you do when the divisor is larger than the first digit of the dividend?

Look at the first two digits together. For example, if you are dividing 435 by 7, the 7 does not go into 4, so you consider 43 instead. Then 7 goes into 43 six times (7 × 6 = 42). Write 6 above the 3, subtract 42 from 43, and continue as usual.

Long Division — Definition, Formula & Examples

Long division is a step-by-step method for dividing a large number (the dividend) by another number (the divisor) to find the quotient and any remainder. It breaks a difficult division problem into a series of easier steps using divide, multiply, subtract, and bring down.

Long division is an algorithm that decomposes the division of a multi-digit dividend by a divisor into sequential partial divisions. At each stage, the procedure determines the largest multiple of the divisor that does not exceed the current partial dividend, records the corresponding digit of the quotient, subtracts the product, and appends the next digit of the dividend to form a new partial dividend. The process terminates when all digits have been processed, yielding a quotient and a non-negative remainder less than the divisor.

Key Formula

\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}

Where:

$\text{Dividend}$ = The number being divided
$\text{Divisor}$ = The number you divide by
$\text{Quotient}$ = The result of the division
$\text{Remainder}$ = The amount left over (always less than the divisor)

How It Works

Long division follows four repeating steps: Divide, Multiply, Subtract, Bring down. First, determine how many times the divisor fits into the leading digit(s) of the dividend. Multiply that number by the divisor and subtract the result from those leading digits. Then bring down the next digit of the dividend and repeat. Continue until there are no more digits to bring down. Any number left over after the final subtraction is the remainder.

Worked Example

Problem: Divide 846 by 3 using long division.

Divide: How many times does 3 go into the first digit, 8? It goes in 2 times (since 3 × 2 = 6). Write 2 above the 8.

8 \div 3 = 2 \text{ R } 2

Multiply and Subtract: Multiply 2 × 3 = 6. Subtract 6 from 8 to get 2.

8 - 6 = 2

Bring Down: Bring down the next digit, 4, to make 24. Now divide: 24 ÷ 3 = 8. Write 8 above the 4. Multiply 8 × 3 = 24 and subtract to get 0.

24 \div 3 = 8, \quad 24 - 24 = 0

Repeat: Bring down the last digit, 6, to make 6. Divide: 6 ÷ 3 = 2. Write 2 above the 6. Multiply 2 × 3 = 6 and subtract to get 0. No remainder.

6 \div 3 = 2, \quad 6 - 6 = 0

Answer: 846 ÷ 3 = 282 with no remainder.

Another Example

Problem: Divide 527 by 4 using long division.

Step 1: 4 does not go into 5 once more than 1 time. 4 × 1 = 4. Write 1 above the 5. Subtract: 5 − 4 = 1.

5 \div 4 = 1 \text{ R } 1

Step 2: Bring down the 2 to make 12. Divide: 12 ÷ 4 = 3. Write 3 above the 2. Subtract: 12 − 12 = 0.

12 \div 4 = 3, \quad 12 - 12 = 0

Step 3: Bring down the 7 to make 7. Divide: 7 ÷ 4 = 1 with remainder 3. Write 1 above the 7. Subtract: 7 − 4 = 3.

7 \div 4 = 1 \text{ R } 3

Answer: 527 ÷ 4 = 131 remainder 3. You can verify: (4 × 131) + 3 = 524 + 3 = 527.

Why It Matters

Long division is a foundational skill practiced throughout elementary and middle school math. It prepares you for dividing polynomials in algebra and working with decimals and fractions. Everyday tasks like splitting a bill evenly, calculating unit prices at a store, or converting units all rely on division fluency.

Common Mistakes

Mistake: Forgetting to write a zero in the quotient when the divisor does not fit into the current partial dividend.

Correction: If the divisor is larger than the number you are working with after bringing down a digit, place a 0 in the quotient and bring down the next digit. Skipping the zero shifts all following digits and gives a wrong answer.

Mistake: Subtracting incorrectly in the middle of the process, which throws off every step that follows.

Correction: Double-check each subtraction before bringing down the next digit. A small subtraction error early on cascades through the entire problem.