---
title: "Division and Multiples"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 32
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/32-division-and-multiples
---

# Chapter 32 — Division and Multiples

Long division learned in [Chapter 23](https://one-course.com/books/math/1/en/chapter/23-division#ch-g4-division) gets two upgrades: two-digit [divisors](#def-g5-division-multiple), and [quotients](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) that continue past the point — $3 \div 4$ finally gets an answer, $0.75$. The chapter ends with [multiples](#def-g5-division-multiple), [divisors](#def-g5-division-multiple) and the first divisibility shortcuts.

## 32.1 Dividing by a two-digit number

**Method 32.1 (Two-digit divisors).**

Same method as before, with one extra skill: guessing how many times the [divisor](#def-g5-division-multiple) fits. Write a small table of [multiples](#def-g5-division-multiple) of the [divisor](#def-g5-division-multiple) first. For $986 \div 23$ ([multiples](#def-g5-division-multiple) of 23: 23, 46, 69, 92, 115, 138, 161, 184, 207):

1. $98$ tens: $23 \times 4 = 92$ fits, $23 \times 5 = 115$ does not: digit $4$ , [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $6$ ;
2. bring down the $6$ : $66$ : $23 \times 2 = 46$ fits: digit $2$ , [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $20$ ;
3. [quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $42$ , [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $20$ . Check: $23 \times 42 + 20 = 966 + 20 = 986$ .

## 32.2 Decimal quotients

**Example 32.2 (The division that refuses to stop at the remainder).**

Share $3$ pizzas among $4$ children: $3 \div 4$ has [quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $0$ and [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $3$ — useless! Continue the division past the point:

1. $3$ units $= 30$ tenths; $30 \div 4$ : $7$ tenths, [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $2$ tenths;
2. $2$ tenths $= 20$ hundredths; $20 \div 4 = 5$ hundredths, [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $0$ .

So $3 \div 4 = 0.75$: each child gets $0.75$ pizza — three [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half), as the [fraction](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) language already knew ([Proposition 31.3](https://one-course.com/books/math/1/en/chapter/31-fractions-as-numbers#prop-g5-fractions-decimal)).

**Method 32.3 (Continuing a division past the point).**

When the units are exhausted and a [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) is left:

1. write the [decimal point](https://one-course.com/books/math/1/en/chapter/25-tenths-and-hundredths#def-g4-decimals-point) in the [quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) ;
2. append a [zero](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero) to the [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) (units become tenths, tenths become hundredths …) and keep dividing;
3. stop when the [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) is $0$ — or when the digits start repeating forever, like $1 \div 3 = 0.333\dots$ : then give a rounded value.

**Example 32.4.**

$27 \div 6$: [quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $4$, [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $3$; point; $30$ tenths $\div\ 6 = 5$: so $27 \div 6 = 4.5$. $22 \div 8$: $2$, [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $6$; then $60 \div 8 = 7$ r $4$; then $40 \div 8 = 5$: so $22 \div 8 = 2.75$.

## 32.3 Multiples and divisors

**Definition 32.5 (Multiple, divisor).**

The *multiples* of $6$ are its table continued forever: $0, 6, 12, 18, 24, \dots$ When $24$ is a multiple of $6$, one also says that $6$ is a *divisor* of $24$, or that $24$ is *divisible* by $6$: the division $24 \div 6$ leaves no [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder).

**Proposition 32.6 (Divisibility shortcuts).**

Without dividing, a number is divisible:

- by $2$ when its last digit is $0$ , $2$ , $4$ , $6$ or $8$ (the [even numbers](https://one-course.com/books/math/1/en/chapter/14-numbers-up-to-10-000#def-g3-numbers-evenodd) );
- by $5$ when its last digit is $0$ or $5$ ;
- by $10$ when its last digit is $0$ ;
- by $3$ when the *[sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def) of its digits* is divisible by $3$ (e.g. $741$ : $7 + 4 + 1 = 12$ , divisible — and indeed $741 = 3 \times 247$ ).

**Proof.** *Admitted at this level.* ∎

![The numbers 1 to 30: multiples of 2 and of 3 make patterns — and the numbers wearing both marks (6, 12, 18, 24, 30) are exactly the multiples of 6.](https://one-course.com/images/onecourse/chapters/math-1/g5-division/fig-19a3bac2302d.svg)

*The numbers $1$ to $30$: [multiples](#def-g5-division-multiple) of $2$ and of $3$ make patterns — and the numbers wearing both marks ($6$, $12$, $18$, $24$, $30$) are exactly the [multiples](#def-g5-division-multiple) of $6$.*

**Example 32.7.**

Is $534$ divisible by $2$? Ends in $4$: yes. By $5$? No. By $3$? $5 + 3 + 4 = 12$: yes. So $534$ is divisible by $6$ as well (by $2$ *and* by $3$) — check: $534 = 6 \times 89$.

## 32.4 Exercises

**Exercise 32.1 ★.**

Compute the long divisions (table of [multiples](#def-g5-division-multiple) first): $851 \div 23$; $704 \div 32$.

**Solution of Exercise 32.1.**

$851 \div 23$: [multiples](#def-g5-division-multiple) of $23$: $23, 46, 69, 92, 115, 138, 161,
184, 207$. Then $85 \div 23$: $3$ times ($69$), [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $16$; bring down the $1$: $161 \div 23 = 7$ exactly. [Quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $37$, [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $0$ ($23 \times 37 = 851$).

$704 \div 32$: $70 \div 32$: $2$ times ($64$), [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $6$; bring down the $4$: $64 \div 32 = 2$. [Quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $22$, [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $0$.

**Exercise 32.2 ★.**

Compute the exact decimal [quotients](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder): $9 \div 2$; $27 \div 4$; $33 \div 5$; $21 \div 8$.

**Solution of Exercise 32.2.**

$9 \div 2 = 4.5$; $27 \div 4 = 6.75$; $33 \div 5 = 6.6$; $21 \div 8 = 2.625$.

**Exercise 32.3 ★.**

Four friends share a $54$ restaurant bill equally. How much does each pay? (Continue past the point.)

**Solution of Exercise 32.3.**

$54 \div 4 = 13.5$: each pays $13.50$.

**Exercise 32.4 ★.**

Compute $10 \div 3$, continuing three digits past the point. What do you observe? Give the value rounded to the hundredth.

**Solution of Exercise 32.4.**

$10 \div 3 = 3.333\dots$: the digit $3$ repeats forever — the division never stops. Rounded to the hundredth: $3.33$.

**Exercise 32.5 ★.**

List: the [multiples](#def-g5-division-multiple) of $7$ up to $70$; the [divisors](#def-g5-division-multiple) of $18$; the [divisors](#def-g5-division-multiple) of $24$.

**Solution of Exercise 32.5.**

[Multiples](#def-g5-division-multiple) of $7$ up to $70$: $7, 14, 21, 28, 35, 42, 49, 56, 63,
70$. [Divisors](#def-g5-division-multiple) of $18$: $1, 2, 3, 6, 9, 18$. [Divisors](#def-g5-division-multiple) of $24$: $1, 2, 3, 4, 6, 8, 12, 24$.

**Exercise 32.6 ★.**

Divisible by $2$? by $5$? by $10$? by $3$? Test each shortcut on: $470$; $735$; $8\,001$; $1\,314$.

**Solution of Exercise 32.6.**

$470$: by $2$ (ends in $0$), by $5$, by $10$; digit [sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def) $11$: not by $3$.

$735$: ends in $5$: by $5$, not by $2$ nor $10$; digit [sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def) $15$: by $3$.

$8\,001$: [odd](https://one-course.com/books/math/1/en/chapter/14-numbers-up-to-10-000#def-g3-numbers-evenodd); digit [sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def) $9$: by $3$ only.

$1\,314$: [even](https://one-course.com/books/math/1/en/chapter/14-numbers-up-to-10-000#def-g3-numbers-evenodd); digit [sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def) $9$: by $2$ and by $3$ (hence by $6$), not by $5$.

**Exercise 32.7 ★.**

Find all the numbers between $60$ and $80$ that are divisible by $3$ *and* by $5$. (Which single divisibility does that combine?)

**Solution of Exercise 32.7.**

Divisible by $3$ and $5$ means divisible by $15$: between $60$ and $80$, the numbers $60$, $75$.

**Exercise 32.8 ★.**

A ribbon of $7.2$ m is cut into $6$ equal pieces. How long is each piece? (Divide a decimal by a whole: share the meters, then the tenths.)

**Solution of Exercise 32.8.**

$7.2 \div 6 = 1.2$: each piece measures $1.2$ m (check: $1.2 \times 6 = 7.2$).

**Exercise 32.9 ★.**

$1\,000$ marbles are packed in bags of $24$. How many full bags, and how many marbles remain? (Two-digit [divisor](#def-g5-division-multiple).)

**Solution of Exercise 32.9.**

$1\,000 = 24 \times 41 + 16$: forty-one full bags, $16$ marbles left.

**Exercise 32.10 ★★.**

Can $5$ friends share $7$ chocolate bars equally, with nothing left? Give each share as a decimal. Same question for $7$ friends sharing $5$ bars — what goes differently?

**Solution of Exercise 32.10.**

$7 \div 5 = 1.4$: each of the $5$ friends gets $1.4$ bars — possible by cutting bars into tenths. For $5 \div 7$: the division gives $0.714285\dots$, which never stops — no exact decimal share exists; only the [fraction](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) $\frac57$ names it exactly.

**Exercise 32.11 ★★.**

Using the [digit-sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def) shortcut, find the smallest digit $d$ that makes $52d$ (a three-digit number ending in $d$) divisible by $3$. Then find all such digits $d$.

**Solution of Exercise 32.11.**

Digit [sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def): $5 + 2 + d = 7 + d$. Divisible by $3$ when $7 + d$ is $9$, $12$ or $15$: $d = 2$, $5$ or $8$. The smallest is $d = 2$ (giving $522 = 3 \times 174$).