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calendar_month Last update: 2025-12-24

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Working with special numbers booklet

calendar_month 2025-12-24

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Unit 1: Numbers
Unit 2: Geometry and measure
Unit 3 : Statistics and probability

🎯 In this topic you will

Use odd and even numbers correctly in problem solving.
Identify and work with factors and multiples of numbers.
Recognise and use square numbers.

🧠 Key Words

even
factor
multiple
odd
square number

Show Definitions

even: A number that can be divided by 2 with no remainder.
factor: A whole number that divides exactly into another number.
multiple: A number obtained by multiplying a given number by a whole number.
odd: A number that cannot be divided by 2 exactly.
square number: A number made by multiplying a whole number by itself.

Working with Special Numbers 🔢

I n this section, we will work with these special numbers.

📘 Worked example

Write these numbers in the correct place on the Venn diagram:
$5,\;6,\;7,\;8,\;10,\;15,\;20$

Answer:

First, find the factors of $30$ and the factors of $40$.

$6$ and $15$ are factors of $30$.
$8$ and $20$ are factors of $40$.
$5$ and $10$ are factors of both $30$ and $40$.

If a number is a factor of both $30$ and $40$, write it where the circles overlap.
If a number is not a factor of $30$ or $40$, write it outside the circles but inside the rectangle.

$7$ is not a factor of $30$ or $40$, so it is written outside the circles.

A factor divides exactly into a number with no remainder.

Check each number by seeing whether it divides $30$ or $40$ exactly.

Numbers that divide both go in the overlap. Numbers that divide only one go in that circle. Numbers that divide neither stay outside.

❓ EXERCISES

1. Mia makes a number using the digits $4$, $5$ and $6$.

The number is even.

The hundreds digit is greater than $4$.

The tens digit is greater than the hundreds digit.

What is Mia’s number? Copy the boxes and write Mia’s number in them.

👀 Show answer

The number is $654$.

2. Here is a Venn diagram for sorting numbers.

Copy and complete the diagram. Write each of these numbers in the correct place.

$10,\;11,\;12,\;13,\;14,\;15,\;16$

👀 Show answer

Multiples of $4$: $12$, $16$.
Multiples of $2$ only: $10$, $14$.
Outside the circles: $11$, $13$, $15$.

3. Here are four number cards.

$30,\;32,\;33,\;35$

Use each number once to make these statements correct.

a. ⬜ is a multiple of $3$.

b. ⬜ is a multiple of $4$.

c. ⬜ is a multiple of $5$.

d. ⬜ is a multiple of $6$.

👀 Show answer

a. $33$
b. $32$
c. $35$
d. $30$

4. Find two square numbers that have a sum of $100$.

👀 Show answer

$36 + 64 = 100$.

5. Copy and complete this sentence.

Every number with a factor of $6$ must also have factors of ⬜, ⬜ and ⬜.

👀 Show answer

The factors are $1$, $2$ and $3$.

6. Pablo writes a $3$-digit number.

All of the digits are odd.

The sum of the digits is $7$.

What is the smallest number Pablo can write? Copy the boxes and write the number in them.

👀 Show answer

The smallest number is $115$.

7. Ingrid says, “All numbers that end in a $4$ are multiples of $4$.”

Is Ingrid correct? Explain how you know.

👀 Show answer

Ingrid is not correct. For example, $14$ ends in $4$ but $14 \div 4$ is not a whole number, so $14$ is not a multiple of $4$.

🧠 Think like a Mathematician

Multiples

You will need these cards:

$0\;1\;2\;3\;4\;5\;6\;7\;8\;9$

Choose a set of multiples, for example multiples of $4$. Now make as many different multiples of $4$ as you can.

You can use each card only once. You may not be able to use all the cards.

Example:

Using the digits, some multiples of $4$ that can be made are: $12$, $40$, $36$, and $8$.

The unused numbers in this example are $5$, $7$, and $9$.

Choose other multiples to investigate.

You will show you are specialising when you find solutions to the problem.

Show possible outcomes

You might find that multiples of $2$ allow more numbers to be formed than multiples of $5$.
Some sets, such as multiples of $9$, are harder because fewer digit combinations work.
Trying different target multiples helps you notice patterns in divisibility.

📘 What we've learned

How to recognise and use odd and even numbers.
How to identify factors and multiples of numbers.
How to work with square numbers such as $1,\;4,\;9,\;16$.
How to classify numbers using Venn diagrams and explain reasoning clearly.

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