Long Division with Remainder — Definition, Formula & Examples
Long division with remainder is a step-by-step method for dividing one number by another when the divisor does not go into the dividend evenly. The leftover amount that cannot be divided further is called the remainder.
Given a dividend and a divisor (where ), long division is an algorithm that produces a quotient and a remainder such that , where .
Key Formula
Where:
- = The dividend (the number being divided)
- = The divisor (the number you divide by)
- = The quotient (the whole-number result)
- = The remainder (the leftover, where 0 ≤ r < b)
How It Works
You work through the digits of the dividend from left to right. At each step, you ask how many times the divisor fits into the current number, write that digit in the quotient, multiply, and subtract. Then you bring down the next digit and repeat. When there are no more digits to bring down, whatever is left over is the remainder.
Worked Example
Problem: Divide 157 by 4 using long division.
Step 1: How many times does 4 go into 15? It fits 3 times. Write 3 above the 5. Multiply and subtract.
Step 2: Bring down the 7 to make 37. How many times does 4 go into 37? It fits 9 times. Multiply and subtract.
Step 3: There are no more digits to bring down. The leftover is 1.
Answer: 157 ÷ 4 = 39 remainder 1
Why It Matters
Long division is essential whenever you split things into equal groups and need to know what's left over — for example, figuring out how many full buses are needed for a field trip and how many students remain. It also builds the foundation for dividing polynomials in algebra.
Common Mistakes
Mistake: Writing a remainder that is larger than or equal to the divisor.
Correction: The remainder must always be less than the divisor. If it is not, the divisor fits in at least one more time — increase your quotient digit and subtract again.