Primary & Middle School Mathematics · Grades 1–9

32Division and Multiples

Long division learned in Chapter 23 gets two upgrades: two-digit divisors, and quotients that continue past the point — $3 \div 4$ finally gets an answer, $0.75$ . The chapter ends with multiples, divisors 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 fits. Write a small table of multiples of the divisor first. For $986 \div 23$ (multiples of 23: 23, 46, 69, 92, 115, 138, 161, 184, 207):

$98$ tens: $23 \times 4 = 92$ fits, $23 \times 5 = 115$ does not: digit $4$ , remainder $6$ ;
bring down the $6$ : $66$ : $23 \times 2 = 46$ fits: digit $2$ , remainder $20$ ;
quotient $42$ , 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 $0$ and remainder $3$ — useless! Continue the division past the point:

$3$ units $= 30$ tenths; $30 \div 4$ : $7$ tenths, remainder $2$ tenths;
$2$ tenths $= 20$ hundredths; $20 \div 4 = 5$ hundredths, remainder $0$ .

So $3 \div 4 = 0.75$ : each child gets $0.75$ pizza — three quarters, as the fraction language already knew (Proposition 31.3).

Method 32.3 (Continuing a division past the point)

When the units are exhausted and a remainder is left:

write the decimal point in the quotient;
append a zero to the remainder (units become tenths, tenths become hundredths …) and keep dividing;
stop when the 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 $4$ , remainder $3$ ; point; $30$ tenths $\div\ 6 = 5$ : so $27 \div 6 = 4.5$ . $22 \div 8$ : $2$ , 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.

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);
by $5$ when its last digit is $0$ or $5$ ;
by $10$ when its last digit is $0$ ;
by $3$ when the sum 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. — 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 $6$ .

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 first): $851 \div 23$ ; $704 \div 32$ .

Solution

Solution of Exercise 32.1.

$851 \div 23$ : multiples of $23$ : $23, 46, 69, 92, 115, 138, 161, 184, 207$ . Then $85 \div 23$ : $3$ times ( $69$ ), remainder $16$ ; bring down the $1$ : $161 \div 23 = 7$ exactly. Quotient $37$ , remainder $0$ ( $23 \times 37 = 851$ ).

$704 \div 32$ : $70 \div 32$ : $2$ times ( $64$ ), remainder $6$ ; bring down the $4$ : $64 \div 32 = 2$ . Quotient $22$ , remainder $0$ .

Exercise 32.2 ★

Compute the exact decimal quotients: $9 \div 2$ ; $27 \div 4$ ; $33 \div 5$ ; $21 \div 8$ .

Solution

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

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

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 of $7$ up to $70$ ; the divisors of $18$ ; the divisors of $24$ .

Solution

Solution of Exercise 32.5.

Multiples of $7$ up to $70$ : $7, 14, 21, 28, 35, 42, 49, 56, 63, 70$ . Divisors of $18$ : $1, 2, 3, 6, 9, 18$ . Divisors 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

Solution of Exercise 32.6.

$470$ : by $2$ (ends in $0$ ), by $5$ , by $10$ ; digit sum $11$ : not by $3$ .

$735$ : ends in $5$ : by $5$ , not by $2$ nor $10$ ; digit sum $15$ : by $3$ .

$8\,001$ : odd; digit sum $9$ : by $3$ only.

$1\,314$ : even; digit sum $9$ : 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

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

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.)

Solution

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

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 $\frac57$ names it exactly.

Exercise 32.11 ★★

Using the digit-sum 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

Solution of Exercise 32.11.

Digit sum: $5 + 2 + d = 7 + d$ . Divisible by $3$ when $7 + d$ is $9$ , $12$ or $15$ : $d = 2$ , $5$ or $8$ . The smallest is $d = 2$ (giving $522 = 3 \times 174$ ).