Topics
3. Multiplication and division
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AustraliaNSW
Stage 3

3.13 Divisibility tests

Lesson
Practice
Lesson

Are you ready?

Being able to remember our  multiplication facts  will help us be able to solve problems in this lesson. Let's try a practice problem.

Examples

Example 1

What is 6 \times 3?

Worked Solution
Create a strategy

We can think of it as 6 groups of 3 like in the array below which has 6 rows of 3 squares.

An array of squares with 6 rows and 3 columns
Apply the idea

There is a total of 18 squares in the array. So:

6\times3=18

Idea summary

There are different ways to solve multiplication, including repeated addition and using arrays. We can also multiply our numbers in a different order, so 3\times4 has the same answer as 4\times3.

Divisibility by 2, 5 or 10

This video looks at how we can tell if a number is divisible by 2,\,5 or 10.

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Examples

Example 2

Are the following numbers divisible by 5?

a

6168

Worked Solution
Create a strategy

We need to check if the last digit of the number is either 5 or 0.

Apply the idea

The last digit of the number is 8, so 6168 is not divisible by 5.

b

3625

Worked Solution
Create a strategy

We need to check if the last digit of the number is either 5 or 0.

Apply the idea

The last digit of the number is 5, so 3625 is divisible by 5.

Idea summary

  • A number is divisible by 2 if it is even.

  • A number is divisible by 5 if it ends in a 0 or 5.

  • A number is divisible by 10 if it ends in a 0.

Divisibility by 3 and 9

This video looks at divisibility by 3 and 9.

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Examples

Example 3

Which of the following numbers are exactly divisible by 3?

A
73\,500
B
73\,495
Worked Solution
Create a strategy

We need to check if the sum of all the digits is divisible by 3.

Apply the idea

Option A:

\displaystyle \text{Sum of digits}\displaystyle =\displaystyle 7+3+5+0+0Add the digits
\displaystyle =\displaystyle 15

Option B:

\displaystyle \text{Sum of digits}\displaystyle =\displaystyle 7+3+4+9+5Add the digits
\displaystyle =\displaystyle 28

15 is divisible by 3, but 28 is not.

So option A: 73\,500 is divisible by 3.

Idea summary

  • A number is divisible by 3 if the sum of its digits is divisible by 3.

  • A number is divisible by 9 if the sum of its digits is divisible by 9.

Divisibility by 4, 6 or 8

This video looks at how we can tell if a number is divisible by 4,\,6 or 8.

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Examples

Example 4

Are the following numbers divisible by 4?

a

58\,784

Worked Solution
Create a strategy

We need to check if the last two digits of the number are 0 or divisible by 4.

Apply the idea

The last two digits of the number is 84, which is divisible by 4.

So, 58\,784 is divisible by 4.

b

372\,938

Worked Solution
Create a strategy

We need to check if the last two digits of the number are 0 or divisible by 4.

Apply the idea

The last two digits of the number is 38, which is not divisible by 4.

So, 372\,938 is not divisible by 4.

Idea summary

  • A number is divisible by 4 if it can be halved twice, or the last 2 digits are divisible by 4.

  • A number is divisible by 6 if it is even and the sum of its digits is divisible by 3.

  • A number is divisible by 8 if it can be halved three times.

Outcomes

MA3-4NA

orders, reads and represents integers of any size and describes properties of whole numbers

MA3-6NA

selects and applies appropriate strategies for multiplication and division, and applies the order of operations to calculations involving more than one operation

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