Number Bases

Lessons

Senior 3 · Mathematics

Number Bases

5 lessons

Lesson 4 of 5

Multiplying and dividing in other bases

25 min

A multiplication table you can learn in a minute

In base ten you learned a hundred multiplication facts. In base five there are only the products of 0 to 4, and in base two just one fact matters: $1 \times 1 = 1$ . Long multiplication and division in any base use the same layouts as in base ten; what changes is how each product is written and carried.

Multiplication facts in base five (each entry written in base five):

$\times$	1	2	3	4
0	0	0	0	0
1	1	2	3	4
2	2	4	11	13
3	3	11	14	22
4	4	13	22	31

For example $4 \times 4 = 16 = 3 \times 5 + 1$ , which is written $31_{5}$ .

Definition

Partial product: in long multiplication, the product of the top number and one digit of the bottom number, written in the place of that digit.

Notation: the partial products are written one under another, each shifted one place further left, and then added.

Example: in $243_5 \times 34_5$ , the partial products are $243_5 \times 4$ and $243_5 \times 30_5$ .

Important

Rule (multiplying by the base). In base $b$ , multiplying a numeral by $10_{b}$ moves every digit one place to the left and puts 0 in the ones place: $243_5 \times 10_5 = 2430_5$ , and $1011_2 \times 10_2 = 10110_2$ .

Why: $10_{b} = b$ , and multiplying by $b$ turns every place value $b^{k}$ into $b^{k+1}$ , the value of the next place on the left. This is why each partial product is shifted.

Multiplying by $10_{b}$ moves every digit one place to the left and puts 0 in the ones place, which multiplies the value by $b$ . This is why each partial product is shifted.

Worked example 1: multiplying by one digit in base five

Work out $243_5 \times 3$ .

	$5^{3}$	$5^{2}$	$5^{1}$	$5^{0}$
carry	1	2	1
		2	4	3
$\times$				3
$=$	1	3	3	4

$3 \times 3 = 9 = 1 \times 5 + 4$ : write 4, carry 1.
$3 \times 4 + 1 = 13 = 2 \times 5 + 3$ : write 3, carry 2.
$3 \times 2 + 2 = 8 = 1 \times 5 + 3$ : write 3, carry 1 into a new column.

So $243_5 \times 3 = 1334_5$ . Check: $73 \times 3 = 219 = 125 + 75 + 15 + 4$ . ✓

Worked example 2: long multiplication in base five

Work out $243_5 \times 34_5$ .

Solution First the partial product for the digit 4:

	$5^{3}$	$5^{2}$	$5^{1}$	$5^{0}$
carry	2	3	2
		2	4	3
$\times$				4
$=$	2	1	3	2

( $4 \times 3 = 12 = 22_5$ : write 2, carry 2; $4 \times 4 + 2 = 18 = 33_5$ : write 3, carry 3; $4 \times 2 + 3 = 11 = 21_5$ : write 1, carry 2.)

The digit 3 stands for $30_{5}$ , so its partial product is $1334_{5}$ (Worked example 1) shifted one place: $13340_{5}$ . Add the partial products in base five:

	$5^{4}$	$5^{3}$	$5^{2}$	$5^{1}$	$5^{0}$
			2	4	3
$\times$				3	4
$243_5 \times 4$		2	1	3	2
$243_5 \times 30_5$	1	3	3	4	0
carry	1	1	1
$=$	2	1	0	2	2

So $243_5 \times 34_5 = 21022_5$ . Check: $73 \times 19 = 1387$ , and $2 \times 625 + 1 \times 125 + 0 + 2 \times 5 + 2 = 1387$ . ✓

Worked example 3: base two, base eight and base sixteen

(a) $1011_2 \times 101_2$ . Each partial product is either a copy of $1011_{2}$ or a row of zeros:

	$2^{5}$	$2^{4}$	$2^{3}$	$2^{2}$	$2^{1}$	$2^{0}$
			1	0	1	1
$\times$				1	0	1
$1011_2 \times 1$			1	0	1	1
$1011_2 \times 0$ , shifted		0	0	0	0
$1011_2 \times 100_2$	1	0	1	1	0	0
$=$	1	1	0	1	1	1

Answer $110111_{2}$ . Check: $11 \times 5 = 55 = 32 + 16 + 4 + 2 + 1$ . ✓

(b) $76_8 \times 5$

	$8^{2}$	$8^{1}$	$8^{0}$
carry	4	3
		7	6
$\times$			5
$=$	4	6	6

$5 \times 6 = 30 = 3 \times 8 + 6$ : write 6, carry 3; $5 \times 7 + 3 = 38 = 4 \times 8 + 6$ : write 6, carry 4. Answer $466_{8}$ . Check: $62 \times 5 = 310 = 256 + 48 + 6$ . ✓

	$16^{2}$	$16^{1}$	$16^{0}$
carry	1	4
		3	B
$\times$			7
$=$	1	9	D

$7 \times \mathrm{B} = 7 \times 11 = 77 = 4 \times 16 + 13$ : write D, carry 4; $7 \times 3 + 4 = 25 = 16 + 9$ : write 9, carry 1. Answer $19\mathrm{D}_{16}$ . Check: $59 \times 7 = 413 = 256 + 144 + 13$ . ✓

Dividing in other bases

There are two safe methods.

Short division in the base. Divide digit by digit from the left, as in base ten. A remainder in one column is worth $b$ times as much in the next column, so multiply it by $b$ (not 10) before adding the next digit.
Through base ten. Convert both numbers to base ten, divide, and convert the quotient back. This is often quicker for long division, and it is always a good check.

Worked example 4: short division

(a) $1334_5 \div 3$ (b) $466_8 \div 5$

Solution

(a) $1 \div 3 = 0$ remainder 1. Carry the 1 into the next column as $1 \times 5 = 5$ : $5 + 3 = 8$ , and $8 \div 3 = 2$ remainder 2. Next: $2 \times 5 + 3 = 13$ , $13 \div 3 = 4$ remainder 1. Last: $1 \times 5 + 4 = 9$ , $9 \div 3 = 3$ remainder 0. Quotient $243_{5}$ , which agrees with Worked example 1. ✓

(b) $4 \div 5 = 0$ remainder 4. Next: $4 \times 8 + 6 = 38$ , $38 \div 5 = 7$ remainder 3. Last: $3 \times 8 + 6 = 30$ , $30 \div 5 = 6$ remainder 0. Quotient $76_{8}$ . ✓

Worked example 5: long division in base two

Work out $110111_2 \div 101_2$ .

Solution In base two each quotient digit is 0 or 1: the divisor either fits or it does not.

$110_2 \ge 101_2$ : write 1; $110_{2} - 101_{2} = 1_{2}$ .
Bring down 1: $11_2 \lt 101_2$ : write 0.
Bring down 1: $111_2 \ge 101_2$ : write 1; $111_{2} - 101_{2} = 10_{2}$ .
Bring down 1: $101_2 \ge 101_2$ : write 1; remainder 0.

Quotient $1011_{2}$ , remainder 0. Check through base ten: $55 \div 5 = 11 = 1011_2$ . ✓

Warning

Change every product into the base before you write it. In base five, $4 \times 4 = 16$ is written $31_{5}$ : write 1 and carry 3. Writing 6 and carrying 1 (the base-ten digits of 16) is the most common mistake in this lesson.

Tip

Keep a base-ten check beside you. Convert the two numbers to base ten at the start. When you finish, convert your answer too: the base-ten product or quotient must match.

JaberTime: Multiplying in base 5 (11 min). Two long multiplications, 432 × 204 and 321 × 24 in base five. Watch how each product is changed into base five before a digit is written and the rest carried.

✦

Practice

Check each answer in base ten. The last three are harder.

Write $4 \times 3$ in base five.
$32_5 \times 4$
$111_2 \times 11_2$
$57_8 \times 6$
$\mathrm{A}7_{16} \times 3$
$1331_5 \div 3$
$101101_2 \div 11_2$
$23_5 \times 14_5$

Show answersHide answers

$12 = 22_{5}$
$233_{5}$ ( $17 \times 4 = 68$ )
$10101_{2}$ ( $7 \times 3 = 21$ )
$432_{8}$ ( $47 \times 6 = 282$ )
$1\mathrm{F}5_{16}$ ( $167 \times 3 = 501$ )
$242_{5}$ ( $216 \div 3 = 72$ )
$1111_{2}$ ( $45 \div 3 = 15$ )
$432_{5}$ ( $13 \times 9 = 117$ )

Sources and credits

4.4: Multiplication and Division in Base Systems, from Contemporary Mathematics by Donna Kirk and contributors, OpenStax, provided by LibreTexts, licensed under CC BY 4.0 (licence read on the page, 30 September 2026). Adapted by Ganzaa.
The base-five multiplication table, the carry tables and the worked examples. Written by Ganzaa.
Video: Multiplying in base 5., JaberTime, embedded from YouTube.

The Rwandan worked examples, the diagrams, the practice items and the quiz questions were written by Ganzaa.

1A multiplication table you can learn in a minute

2Dividing in other bases

A multiplication table you can learn in a minute

Dividing in other bases