Long Division: The Standard Algorithm

Four steps, repeated

Long division finds the answer one digit at a time. Each digit takes the same four steps.

Step	What you do
divide	how many times does the divisor fit?
multiply	that many times the divisor
subtract	take it away from what you had
bring down	fetch the next digit

Then repeat, until there are no digits left to bring down.

Working an example

$$372 \div 4$$

Does $4$ fit into $3$? No, so take the first two digits: $37$.

Divide: $4$ fits into $37$ nine times. Multiply: $9 \times 4 = 36$. Subtract: $37 - 36 = 1$. Bring down: the $2$, giving $12$.

Repeat with $12$:

Divide: $4$ fits into $12$ three times. Multiply: $3 \times 4 = 12$. Subtract: $12 - 12 = 0$. Bring down: nothing left.

$$372 \div 4 = 93$$ $$\begin{array}{r} 93 \\ 4\,\overline{)372} \\ \end{array}$$

Why the four steps work

The steps are not arbitrary. Each cycle answers one question: how many of the divisor fit into what remains, at this place value?

The first cycle above found $9$, and that $9$ sits in the tens column — so it really means ninety. Checking: $90 \times 4 = 360$, which is most of the $372$. The second cycle deals with the $12$ that was left over.

So long division is repeated subtraction, organized by place value. Taking away $4$ from $372$ ninety-three times would give the same answer far more slowly.

Remainders

The divisor often does not fit a whole number of times.

$$47 \div 5$$

$5$ fits into $47$ nine times, $9 \times 5 = 45$, and $47 - 45 = 2$.

$$47 \div 5 = 9 \text{ remainder } 2$$

A remainder can be written three ways, and which you want depends on the question:

Form	Result	Use when
remainder	$9$ r $2$	the leftover cannot be split
fraction	$9\tfrac{2}{5}$	an exact answer is wanted
decimal	$9.4$	a measurement is wanted

How many full boxes of 5? — the remainder form. How much rope each? — the fraction or decimal.

Zeros in the quotient

If the divisor does not fit, write a zero and carry on.

$$618 \div 6$$

$6$ fits into $6$ once. Subtract, bring down the $1$: $6$ does not fit into $1$, so the next digit of the answer is $0$. Bring down the $8$: $6$ fits into $18$ three times.

$$618 \div 6 = 103$$

Skipping the zero would give $13$, which is ten times too small. A zero in the quotient is a real digit, holding the place, and leaving it out shifts everything after it.

Checking

Multiplication undoes division, so:

$$\text{quotient} \times \text{divisor} + \text{remainder} = \text{original}$$

Checking $372 \div 4 = 93$:

$$93 \times 4 = 372 \; ✓$$

Checking $47 \div 5 = 9$ r $2$:

$$9 \times 5 + 2 = 45 + 2 = 47 \; ✓$$

This check is worth doing every time. It is quick, it uses a different operation, and it catches a wrong digit anywhere in the answer.

A two-digit divisor

The method is unchanged. Only the “how many times” step takes more thought.

$$966 \div 23$$

$23$ does not fit into $9$, so start with $96$. Estimating helps: $23 \times 4 = 92$, so four times.

Subtract: $96 - 92 = 4$. Bring down the $6$: $46$.

$23 \times 2 = 46$ exactly, so the next digit is $2$.

$$966 \div 23 = 42$$

Check: $42 \times 23 = 966$ ✓

Worked examples

Common mistakes

Practice problems