Long Division Formula

The Formula

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

When to use: Long division is like distributing items into groups one place value at a time. If you have 156 stickers to share among 12 friends, you first figure out how many groups of 12 fit in 156 by working from the biggest place value down: how many 12s in 15? Then bring down the next digit and repeat.

Quick Example

156 \div 12 = 13 \quad \text{because } 12 \times 13 = 156

Notation

The division bracket \overline{)\phantom{0}} places the quotient on top, the dividend inside, and the divisor outside

What This Formula Means

A step-by-step algorithm for dividing a multi-digit number (dividend) by another number (divisor), producing a quotient and possibly a remainder.

Long division is like distributing items into groups one place value at a time. If you have 156 stickers to share among 12 friends, you first figure out how many groups of 12 fit in 156 by working from the biggest place value down: how many 12s in 15? Then bring down the next digit and repeat.

Formal View

a = bq + r where a is the dividend, b is the divisor, q = \lfloor a/b \rfloor is the quotient, and 0 \leq r < b is the remainder (Division Algorithm)

Worked Examples

Example 1

easy
Compute 846 \div 6.

Solution

  1. 1
    Divide 8 by 6: quotient digit 1, remainder 8 - 6 = 2. Bring down 4 to get 24.
  2. 2
    Divide 24 by 6: quotient digit 4, remainder 0. Bring down 6 to get 6.
  3. 3
    Divide 6 by 6: quotient digit 1, remainder 0.
  4. 4
    Result: 846 \div 6 = 141.

Answer

141
Long division works digit by digit from left to right: divide, multiply, subtract, bring down, and repeat.

Example 2

medium
Compute 1573 \div 11.

Common Mistakes

  • Forgetting to bring down the next digit before continuing
  • Placing a quotient digit in the wrong position (misalignment)
  • Not writing a zero in the quotient when the divisor doesn't fit into a partial dividend (e.g., 408 \div 4 = 102, not 12)

Why This Formula Matters

Long division is essential for working with large numbers and is the basis for dividing decimals and polynomials later.

Frequently Asked Questions

What is the Long Division formula?

A step-by-step algorithm for dividing a multi-digit number (dividend) by another number (divisor), producing a quotient and possibly a remainder.

How do you use the Long Division formula?

Long division is like distributing items into groups one place value at a time. If you have 156 stickers to share among 12 friends, you first figure out how many groups of 12 fit in 156 by working from the biggest place value down: how many 12s in 15? Then bring down the next digit and repeat.

What do the symbols mean in the Long Division formula?

The division bracket \overline{)\phantom{0}} places the quotient on top, the dividend inside, and the divisor outside

Why is the Long Division formula important in Math?

Long division is essential for working with large numbers and is the basis for dividing decimals and polynomials later.

What do students get wrong about Long Division?

Estimating how many times the divisor fits into each partial dividend, especially with two-digit divisors.

What should I learn before the Long Division formula?

Before studying the Long Division formula, you should understand: division, subtraction, multiplication.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Polynomial Long Division: Step-by-Step Method with Examples →
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