Divide Whole Numbers

Use division notation

So far we have explored addition, subtraction, and multiplication. Now let’s consider division. Suppose you have $12$ cookies and want to package them in bags with $4$ cookies in each bag. How many bags would you need? You might put $4$ cookies in the first bag, $4$ in the second, and so on until you run out — filling $3$ bags. In other words, starting with $12$ cookies, you subtract $4$ at a time. Division is a way to represent repeated subtraction, just as multiplication represents repeated addition. Instead of subtracting $4$ repeatedly, we write

We read this as twelve divided by four, and the result is the quotient of $12$ and $4$. The quotient is $3$ because we can subtract $4$ from $12$ exactly $3$ times. The number being divided is the dividend, and the number dividing it is the divisor. Here $12$ is the dividend and $4$ is the divisor.

Division can be written several ways — with the division sign, as a fraction, with a slash, or with the long division bracket:

In each case, $12$ is the dividend and $4$ is the divisor. Division is performed on two numbers at a time; when translating between notation and words, look for the words of and and to identify the numbers.

Example. Translate from math notation to words: (a) $64 \div 8$ (b) $\frac{42}{7}$ (c) $4\overline{)28}$.

(a) Sixty-four divided by eight; the result is the quotient of sixty-four and eight. (b) Forty-two divided by seven; the result is the quotient of forty-two and seven. (c) Twenty-eight divided by four; the result is the quotient of twenty-eight and four.

Model division of whole numbers

As with multiplication, we model division using counters — here, division organizes items into equal groups. To find the quotient $24 \div 8$, we want to know how many groups of $8$ are in $24$. Model the dividend by starting with $24$ counters, then form groups of $8$ counters — the divisor tells us how many counters go in each group. Counting the groups, there are $3$:

Divide whole numbers

We said that addition and subtraction are inverse operations because one undoes the other. Similarly, division is the inverse operation of multiplication. We know $12 \div 4 = 3$ because $3 \cdot 4 = 12$. Knowing the multiplication facts is very important when doing division — and we check a division by multiplying the quotient by the divisor to see if it equals the dividend. We know $24 \div 8 = 3$ is correct because $3 \cdot 8 = 24$.

For example, $42 \div 6 = 7$ (check: $7 \cdot 6 = 42$ ✓), $\frac{72}{9} = 8$ (check: $8 \cdot 9 = 72$ ✓), and $63 \div 7 = 9$ (check: $9 \cdot 7 = 63$ ✓).

What is the quotient when you divide a number by itself?

Division Properties of One: any number (except $0$) divided by itself is one, $a \div a = 1$; and any number divided by one is the same number, $a \div 1 = a$. So $11 \div 11 = 1$, $\frac{19}{1} = 19$, and $1\overline{)7}$ gives $7$.

What about dividing with zero? Suppose we have $0 and want to divide it among $3$ people — each person would get $0. Zero divided by any number is $0$. But now suppose we want to divide $10 by $0$: we would need a number that we multiply by $0$ to get $10$, and that cannot happen, because $0$ times any number is $0$.

Division Properties of Zero: zero divided by any number is zero, $0 \div a = 0$; dividing a number by zero is undefined, $a \div 0$ has no answer. Another way to see this: division is repeated subtraction, and subtracting $0$ from $10$ will never change the total, so we never get an answer.

When the divisor or the dividend has more than one digit, it is usually easier to use the bracket notation — a process called long division. Let’s divide $78$ by $3$. The divisor $3$ goes into $7$ (the first digit of the dividend) two times, since $2 \times 3 = 6$: write the $2$ above the $7$, write the product $6$ under the $7$, and subtract, leaving $1$. Bring down the next digit, $8$, making $18$; the divisor $3$ goes into $18$ six times, so write $6$ in the quotient above the $8$. Multiply $6 \times 3 = 18$ and subtract — nothing remains, and there are no more digits to bring down, so the division is finished:

Check by multiplying the quotient times the divisor: $26 \times 3 = 78$. ✓

To divide whole numbers:

1. Divide the first digit of the dividend by the divisor. If the divisor is larger than the first digit of the dividend, divide the first two digits of the dividend by the divisor, and so on.
2. Write the quotient above the dividend.
3. Multiply the quotient by the divisor and write the product under the dividend.
4. Subtract that product from the dividend.
5. Bring down the next digit of the dividend.
6. Repeat from step 1 until there are no more digits in the dividend to bring down.
7. Check by multiplying the quotient times the divisor.

Example. Divide $2{,}596 \div 4$. Since $4$ does not go into $2$, use the first two digits: $4$ goes into $25$ six times. Write the $6$, multiply $6 \cdot 4 = 24$, subtract to leave $1$, and bring down the $9$. There are $4$ fours in $19$; write the $4$, subtract $16$ from $19$ to leave $3$, and bring down the $6$. There are $9$ fours in $36$, with nothing left over. So $2{,}596 \div 4 = 649$. Check: $649 \times 4 = 2{,}596$. ✓

Example. Divide $4{,}506 \div 6$. $6$ doesn’t go into $4$, so start with $45$: there are $7$ sixes in $45$, with $3$ left over. Bring down the $0$, making $30$; there are $5$ sixes in $30$, exactly. Bring down the $6$; there is $1$ six in $6$, exactly. So $4{,}506 \div 6 = 751$. Check: $751 \times 6 = 4{,}506$. ✓

Example. Divide $7{,}263 \div 9$. Since $9$ doesn’t go into $7$, start with $72$: there are $8$ nines in $72$, exactly. Bring down the $6$; there are $0$ nines in $6$, so write a $0$ in the quotient. Bring down the $3$, making $63$; there are $7$ nines in $63$. So $7{,}263 \div 9 = 807$. Check: $807 \times 9 = 7{,}263$. ✓ (Keep an eye out for zeros in the quotient — they hold their place just like any other digit.)

So far all our divisions have worked out evenly, but that isn’t always so. Suppose there were $28$ cookies to put in bags of $8$: there would be $3$ groups of eight, with $4$ cookies left over. We call the $4$ the remainder, and show it by writing R4 next to the $3$ — the quotient is $3$ R$4$. To check a division with a remainder, multiply the quotient by the divisor, then add the remainder: $3 \times 8 = 24$, and $24 + 4 = 28$. ✓

Example. Divide $1{,}439 \div 4$. Working through the long division: $4$ into $14$ goes $3$ times leaving $2$; bring down the $3$ to make $23$, which holds $5$ fours, leaving $3$; bring down the $9$ to make $39$, which holds $9$ fours, leaving $3$. There are no more digits to bring down, so $1{,}439 \div 4$ is $359$ with a remainder of $3$. Check: $359 \cdot 4 = 1{,}436$, and $1{,}436 + 3 = 1{,}439$. ✓

Divisors can have more than one digit too. Dividing $1{,}461 \div 13$ the same way gives $112$ R$5$ (check: $112 \cdot 13 = 1{,}456$, plus $5$ is $1{,}461$ ✓). Dividing $74{,}521 \div 241$ gives $309$ R$52$. Sometimes it might not be obvious how many times the divisor goes into digits of the dividend — we have to guess and check to find the greatest number that goes in without exceeding them.

Translate word phrases to math notation

Some of the words that indicate division:

Example. Translate and simplify: the quotient of $51$ and $17$. The word quotient tells us to divide: $51 \div 17 = 3$. We could just as correctly have written it $17\overline{)51}$ or $\frac{51}{17}$.

Divide whole numbers in applications

We use the same strategy as in previous sections: determine what we are looking for, write a phrase, translate it to math notation, simplify, and answer with a complete sentence.

Example. Cecelia bought a $160$-ounce box of oatmeal at the big box store. She wants to divide it into $8$-ounce servings, one bag for each work day. How many servings will she get from the big box? The phrase is $160$ ounces divided by $8$ ounces, which translates to $160 \div 8 = 20$. Cecelia will get $20$ servings from the big box.

Key terms

quotient — the result of dividing one number by another. dividend — the number being divided. divisor — the number dividing the dividend. remainder — the amount left over when a division does not come out evenly. undefined — having no answer; dividing a number by zero is undefined. long division — the digit-by-digit process of dividing using the bracket notation.

This section is adapted from Prealgebra 2e, Section 1.5: Divide Whole Numbers by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: condensed prose, described the cookie and counter models and the long-division worked columns in prose, and converted practice problems (“Try Its”) into interactive exercises with instant feedback.