---
title: "Division"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 23
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/23-division
---

# Chapter 23 — Division

Grade 3 divided with the tables and small [remainders](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) ([Chapter 17](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#ch-g3-division)); this chapter installs the full written method — long division — which handles any number, digit group by digit group.

## 23.1 The long division method

**Method 23.1 (Long division by a one-digit number).**

To divide $749$ by $6$:

1. take the leftmost digit, $7$ (hundreds): $6$ goes $1$ time into $7$ ; write $1$ in the [quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) , subtract $6$ , [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $1$ ;
2. *bring down* the next digit, $4$ : now divide $14$ (tens): $6 \times 2 = 12$ ; write $2$ , [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $2$ ;
3. bring down the $9$ : divide $29$ : $6 \times 4 = 24$ ; write $4$ , [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $5$ ;
4. no digit left: the [quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) is $124$ , the [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $5$ .

Check: $6 \times 124 + 5 = 744 + 5 = 749$, and $5 < 6$.

![The long division of 749 by 6, written in full: each subtraction appears, every digit stays in its column, and the gray arrows show the digits being brought down.](https://one-course.com/images/onecourse/chapters/math-1/g4-division/fig-b130aec5cdd4.svg)

*The long division of $749$ by $6$, written in full: each subtraction appears, every digit stays in its column, and the gray arrows show the digits being brought down.*

**Example 23.2 (A zero in the quotient).**

Divide $618$ by $3$: hundreds, $6 \div 3 = 2$; tens, bring down the $1$: $3$ goes $0$ times into $1$ — *write the* $0$! — [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $1$; units, bring down the $8$: $18 \div 3 = 6$. [Quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $206$, [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $0$. Forgetting the middle [zero](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero) (writing $26$) is the classic mistake; the check $3 \times 26 = 78 \neq 618$ catches it immediately.

**Example 23.3 (Estimating the quotient’s size).**

Before dividing $749$ by $6$, frame the answer: $6 \times 100 = 600$ and $6 \times 200 = 1\,200$, so the [quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) is between $100$ and $200$ — it will have three digits. This one-line estimate prevents most misplaced-digit errors.

## 23.2 Choosing what the question asks

**Method 23.4 (Quotient, remainder, or quotient plus one).**

As in [Method 17.5](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#met-g3-division-interpret), the story decides:

1. “how many full …” or “how many each”: the [quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) ;
2. “how many left”: the [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) ;
3. “how many containers needed for all”: the [quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) , plus one if the [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) is not [zero](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero) .

**Example 23.5.**

$530$ books must be packed in boxes of $8$.

$$
530 = 8 \times 66 + 2 .
$$

How many *full* boxes? $66$. How many books are not in a full box? $2$. How many boxes to pack *all* the books? $67$. Three questions, one division.

**Example 23.6 (Fair sharing of money).**

Four friends share the cost of a $92$ gift equally: $92 \div 4 = 23$ exactly ([remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $0$). Each pays $23$. When the [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) is [zero](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero), the division “comes out even” and the sharing is perfectly fair.

## 23.3 Exercises

**Exercise 23.1 ★.**

Frame each [quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) as in [Example 23.3](#ex-g4-division-size) (between which two “[round](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round)” numbers?), then compute the long division: $96 \div 4$; $87 \div 5$.

**Solution of Exercise 23.1.**

$96 \div 4$: between $4 \times 20 = 80$ and $4 \times 30 = 120$, so the [quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) has two digits; long division gives $24$, [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $0$.

$87 \div 5$: between $5 \times 10 = 50$ and $5 \times 20 = 100$; [quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $17$, [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $2$ ($87 = 5 \times 17 + 2$).

**Exercise 23.2 ★.**

Compute the long divisions and check each one: $672 \div 4$; $925 \div 7$; $804 \div 6$.

**Solution of Exercise 23.2.**

$672 \div 4 = 168$, [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $0$ (check: $4 \times 168 = 672$).

$925 \div 7 = 132$, [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $1$ (check: $7 \times 132 + 1 = 925$).

$804 \div 6 = 134$, [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $0$ (check: $6 \times 134 = 804$).

**Exercise 23.3 ★.**

Careful with the [zero](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero) (see [Example 23.2](#ex-g4-division-zero)): $816 \div 4$; $420 \div 7$; $2\,512 \div 5$.

**Solution of Exercise 23.3.**

$816 \div 4 = 204$ (the tens step is $1 \div 4$: [zero](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero) times — write the $0$).

$420 \div 7 = 60$.

$2\,512 \div 5 = 502$, [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $2$ (check: $5 \times 502 + 2 = 2\,512$).

**Exercise 23.4 ★.**

Write the check equality ($a = b \times q + r$) for: $85 \div 9$; $1\,000 \div 3$.

**Solution of Exercise 23.4.**

$85 = 9 \times 9 + 4$ ([quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $9$, [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $4$).

$1\,000 = 3 \times 333 + 1$.

**Exercise 23.5 ★.**

$156$ eggs are put in boxes of $6$. How many boxes are filled?

**Solution of Exercise 23.5.**

$156 \div 6 = 26$: twenty-six boxes are filled, none left over.

**Exercise 23.6 ★.**

A ribbon of $250$ cm is cut into pieces of $8$ cm. How many pieces, and what length is left over?

**Solution of Exercise 23.6.**

$250 = 8 \times 31 + 2$: thirty-one pieces, and $2$ cm of ribbon left.

**Exercise 23.7 ★.**

$375$ students go to a show in buses of $52$ seats. How many buses are needed? (Which case of [Method 23.4](#met-g4-division-interpret) is this? Divide by $52$ using multiples: $52 \times 7 = 364$.)

**Solution of Exercise 23.7.**

$375 = 52 \times 7 + 11$: seven buses carry $364$ students, $11$ remain — the “plus one” case: $8$ buses are needed.

**Exercise 23.8 ★.**

Three friends share $234$ marbles equally. How many marbles each? Check with a multiplication.

**Solution of Exercise 23.8.**

$234 \div 3 = 78$: each gets $78$ marbles. Check: $3 \times 78 = 234$.

**Exercise 23.9 ★.**

Find the dividend: the division by $7$ gives [quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $58$ and [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $3$.

**Solution of Exercise 23.9.**

Dividend $= 7 \times 58 + 3 = 406 + 3 = 409$.

**Exercise 23.10 ★★.**

A librarian shelves $438$ books, $9$ per shelf. Shelves come in bookcases of $6$ shelves. How many shelves are needed? How many bookcases must be bought? (Two divisions, both with a “plus one” question.)

**Solution of Exercise 23.10.**

Shelves: $438 = 9 \times 48 + 6$: $48$ full shelves and $6$ books more — $49$ shelves are needed. Bookcases: $49 = 6 \times 8 + 1$: eight full bookcases and one extra shelf — $9$ bookcases must be bought.

**Exercise 23.11 ★★.**

In a division by $8$, the [quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) equals the [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder). What are the possible dividends? List them all. (The [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) must stay below $8$.)

**Solution of Exercise 23.11.**

[Quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $=$ [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $= r$ with $r < 8$, so $r$ is $0, 1, \dots,
7$ and the dividend is $8r + r = 9r$: the possible dividends are $0$, $9$, $18$, $27$, $36$, $45$, $54$, $63$.