1.1

Primes, Prime Factorisation and Index Notation

We learnt about multiples and factors previously.

AMultiples

We can find the multiples of a whole number by multiplying it by another positive whole number.

Example:

The multiples of 3 are $3 \times 1, \; 3 \times 2, \; 3 \times 3, \; 3 \times 4, \; 3 \times 5, \; \ldots$
i.e., $\;3, \;6, \;9, \;12, \;15, \;\ldots$

BFactors

0, 1, 2, 3, 4, ... are whole numbers. A whole number greater than 1 can be expressed as a product of two whole numbers.

Example:

$$\begin{aligned} 12 &= 1 \times 12 \\ &= 2 \times 6 \\ &= 3 \times 4 \end{aligned}$$

1, 2, 3, 4, 6 and 12 are called the factors of 12 and 12 is divisible by each of its factors.

Do you see the link between multiples and factors?

A whole number greater than 1 is divisible by each of its factors, that is, it can be divided by any of its factors without leaving a remainder.

THINK²

(a)(i) Is 5 a factor of 10? Why?
(a)(ii) Is 3 a factor of 10? Why?
(b) Can a whole number greater than 10 be a factor of 10? Give your reasons.

EN (a)(i) Yes. Because $10 = 5 \times 2$, 10 is divisible by 5 with no remainder.
(a)(ii) No. $10 \div 3 = 3$ remainder 1, so 10 is not divisible by 3.
(b) No. A factor of 10 must divide 10 exactly, but any whole number greater than 10 produces a quotient less than 1, so it cannot be a factor of 10.

⏱ 仔细阅读英文版，30 秒后可查看中文版

中文 (a)(i) 是。因为 $10 = 5 \times 2$，10 能被 5 整除，余数为 0。
(a)(ii) 不是。$10 \div 3 = 3$ 余 1，10 不能被 3 整除。
(b) 不能。10 的因数必须能整除 10，但任何大于 10 的整数除 10 都得不到整数商，所以不可能是 10 的因数。

看完答案，自评：

CPrime Numbers and Composite Numbers

Activity 1

Objective: To classify whole numbers based on the number of factors.

(a) Complete the table by listing the factors of 2 to 12. Some examples are given below.

Number	Product(s)	Factors
2	$2 = 1 \times 2$	1, 2
3
4	$4 = 1 \times 4$ $= 2 \times 2$	1, 2, 4
5
6
7
8
9	$9 = 1 \times 9$ $= 3 \times 3$	1, 3, 9
10
11
12	$12 = 1 \times 12$ $= 2 \times 6$ $= 3 \times 4$	1, 2, 3, 4, 6, 12

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(b) Complete the table by grouping the numbers 2 to 12 according to the number of factors they have. 2 and 4 are done for you.

Number of factors	Numbers
Two factors	2,
More than two factors	4,

Look at the numbers with only two factors. What do you observe about the factors of these numbers?

EN Each of these numbers (2, 3, 5, 7, 11) has only two factors: 1 and the number itself. No other whole number can divide them exactly. This property — exactly two factors, namely 1 and the number itself — is the defining feature of a prime number.

⏱ 仔细阅读英文版，30 秒后可查看中文版

中文这些数（2, 3, 5, 7, 11）只有两个因数 — 1 和它本身。没有别的整数能整除它们。这个特性——恰好有两个因数（1 和它自己）——正是质数（prime number）的定义。

看完答案，自评：

From Activity 1, we observe the following.

A whole number greater than 1 can be classified either as a prime number or a composite number.
A prime number has only two factors, 1 and itself.
A composite number has more than two factors.

Note that 0 and 1 are neither prime numbers nor composite numbers.

THINK²

Explain why 0 and 1 are not prime numbers.

EN By definition, a prime number must be greater than 1 and have exactly two distinct factors.
· 0 is not greater than 1, and every non-zero integer divides 0, so 0 has infinitely many factors.
· 1 has only one factor (1 itself), not two distinct factors.

⏱ 仔细阅读英文版，30 秒后可查看中文版

中文定义要求：质数必须 大于 1、且有 恰好两个不同的因数。
· 0 不大于 1，且任何非零整数都能整除 0，所以 0 有无穷多个因数。
· 1 只有一个因数（就是 1 自己），没有"两个不同的因数"。

看完答案，自评：
Can a whole number be both a prime number and a composite number? Why?

EN No. A prime number has exactly two factors; a composite number has more than two factors. The two definitions are mutually exclusive on the number of factors, so the same whole number cannot be both at the same time.

⏱ 仔细阅读英文版，30 秒后可查看中文版

中文 不能。质数有 恰好两个 因数；合数有 多于两个 因数。两者在因数个数上互相排斥，所以同一个整数不能既是质数又是合数。

看完答案，自评：

To determine whether a number is a prime, we can check if the number is divisible by all the prime numbers smaller than itself.

If a number (e.g., 90) is divisible by a composite number (e.g., 18), then it is divisible by all the prime numbers which are the factors (e.g., 1, 2, 3, 6, 9, 18) of this composite number.

WORKED EXAMPLE 1

Determine whether each of the following is a prime or a composite number.
(a) 143 (b) 29

SOLUTION

(a) Check if 143 is divisible by any prime number smaller than 143, such as 2, 3, 5, 7, 11, ...

$143 \div 11 = 13$
$\therefore$ 143 is a composite number because it has more than two factors.

The factors of 143 are 1, 11, 13 and 143.

(b) Check if 29 is divisible by any prime number smaller than 29.

29 is not divisible by any prime number smaller than itself.
$\therefore$ 29 is a prime number because it has only two factors, 1 and itself.

TRY IT YOURSELF 1

Determine whether each of the following is a prime or a composite number.

(a) 234 → (b) 171 → (c) 31 →

THINK²

Show that the following numbers are composite numbers.
(a) 135 (b) 485 (c) 920 (d) 2320
(e) 72 (f) 134 (g) 836 (h) 7128
Using the results in 1, how can we make use of the last digit of a number to show that it is a composite number?

EN If a whole number has a last digit of 0, 2, 4, 5, 6 or 8, then it is composite (provided the number itself is greater than the digit). Numbers ending in 0, 2, 4, 6, 8 are divisible by 2; numbers ending in 0 or 5 are divisible by 5. Either way, they have factors other than 1 and themselves, hence are composite.

⏱ 仔细阅读英文版，30 秒后可查看中文版

中文如果一个整数的末位数字是 0, 2, 4, 5, 6 或 8，那它（只要本身大于这个末位数）就是合数。末位为 0, 2, 4, 6, 8 的能被 2 整除；末位为 0 或 5 的能被 5 整除。这意味着它除了 1 和自己之外还有别的因数，因此是合数。

看完答案，自评：

DPrime Factorisation

Let us consider the composite number 30. Expressing 30 as the product of its prime factors (factors which are prime numbers), we have:

We say that 2, 3 and 5 are the prime factors of 30.

Activity 2

Objective: To express a whole number greater than 1 as a product of its prime factors.

(a) Copy the following and fill in the blanks. The first one has been done for you as an example.

(i)

(ii)

(b) Hence express 10 and 28 as a product of their prime factors. The first one has been done for you as an example.

(i) $10 = 2 \times 5$

(ii) $28 = $
(a) Copy and complete the factor trees.

Factor Tree 1

Factor Tree 2

Factor Tree 3

(b) Using each of the factor trees, write down 280 as the product of its prime factors.

Factor Tree 1: $280 = $

Factor Tree 2: $280 = $

Factor Tree 3: $280 = $

(c) What can you say about the products in question 2(b)?

EN All three products are the same. No matter which two factors we start with at the top level, the prime factorisation always reduces to $2 \times 2 \times 2 \times 5 \times 7$. The order of the prime factors may differ, but the multiset of primes is identical — the prime factorisation of 280 is unique.

⏱ 仔细阅读英文版，30 秒后可查看中文版

中文 三个乘积相同。不管最上层选哪两个因数开始分解，最后都会得到 $2 \times 2 \times 2 \times 5 \times 7$。质因数的写出顺序可能不同，但这一组质因数本身完全一样——280 的质因数分解是唯一的。

看完答案，自评：

From Activity 2, we can see there is more than one way to begin the process of expressing a number as the product of its prime factors such as:

$280 = 10 \times 28, \quad 280 = 2 \times 140, \quad 280 = 4 \times 70, \;\ldots$

Whichever expression we choose to start with, we get the same product of prime factors for 280.

$280 = 2 \times 2 \times 2 \times 5 \times 7$

More importantly, rearranging the order of these prime factors also gives the same value, 280.

Every whole number greater than 1 is either a prime number or it can be expressed as a unique product of its prime factors.
The process of expressing a composite number as a product of its prime factors is called prime factorisation.

WORKED EXAMPLE 2

Express 126 as the product of its prime factors.

SOLUTION

Method 1: Factor Tree

We can build a factor tree as follows:

STEP 1

Write down two numbers whose product is 126, e.g., $126 = 2 \times 63$.

STEP 2

Continue to factorise 2 and 63, if possible.

STEP 3

Stop when the last row shows the prime factors. The product of all the prime factors in the tree is the prime factorisation of the given number.

Written as the product of its prime factors,
$126 = 2 \times 3 \times 3 \times 7$.

Method 2: Successive Short Division

The prime factorisation is the product of all the divisors.
$\therefore 126 = 2 \times 3 \times 3 \times 7$

TRY IT YOURSELF 2

Express 585 as the product of its prime factors using
(a) a factor tree, (b) the successive short division method.

Answer: $585 = $

THINK²

(a) If you are asked to express 360 as the product of its prime factors using the successive short division method, which number will you start with? Why?
(b) Do you think it is a good strategy to start the division process using the smallest prime factor? Why?

看完答案，自评：

EIndex Notation

In Activity 2, we obtained $280 = 2 \times 2 \times 2 \times 5 \times 7$. We see that there is a product of three '2's. We can use notation to 'shorten' this as $2^3$. Thus, we have $280 = 2^3 \times 5 \times 7$.

$2 \times 2 \times 2 = 2^3$, which is read as '2 to the power of 3' or '2 cubed'.

This notation is called index notation. The number 2 is the base and the number 3 is the index. The index shows the number of times the base is multiplied by itself.

Similarly,
$2 \times 2 = 2^2$, which is read as '2 to the power of 2' or '2 squared' and
$2 \times 2 \times 2 \times 2 = 2^4$, which is read as '2 to the power of 4'.

We can use index notation to express the prime factorisation of a number in a more concise manner.

WORKED EXAMPLE 3

Express 720 as the product of its prime factors in index notation.

SOLUTION

Method 1: Factor Tree

$\therefore 720 = 2 \times 5 \times 3 \times 3 \times 2 \times 2 \times 2$
$\quad\quad\;\; = 2^4 \times 3^2 \times 5 \quad$ (index notation)

Method 2: Successive Short Division

$\therefore 720 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5$
$\quad\quad\;\; = 2^4 \times 3^2 \times 5 \quad$ (index notation)

TRY IT YOURSELF 3

Express 702 as the product of its prime factors in index notation.

Answer: $702 = $

PRACTICE EXERCISE 1.1

Basic Mastery

1 Determine whether the following numbers are prime numbers.
(a) 85 → (b) 83 →
(c) 103 → (d) 323 →

填 Y（是质数）或 N（不是）

2 Express the following in index notation.
(a) $8 \times 8 \times 8 = $
(b) $3 \times 3 \times 3 \times 3 \times 3 = $
(c) $7 \times 7 \times 9 = $
(d) $4 \times 4 \times 6 \times 6 = $
(e) $2 \times 3 \times 11 \times 11 \times 11 = $
(f) $5 \times 5 \times 13 \times 5 \times 13 \times 37 = $

3 Find the prime factors of the following numbers using the factor tree method. Write each number as a product of its prime factors in index notation.

(a) 54

(b) 72

(a) $54 = $
(b) $72 = $

4 Find the prime factors of the following using successive short division. Write each as a product of its prime factors in index notation.

(a) 784

$784 = $

(b) 4851

$4851 = $

$6125 = $

5 Express each of the following numbers as a product of its prime factors in index notation.
(a) $180 = $
(b) $616 = $
(c) $735 = $
(d) $1350 = $

Intermediate

6 Find:
(a) the smallest prime factor of 377 →
(b) the largest multiple of 17 that is less than 1000 →
(c) the smallest multiple of 19 that is greater than 500 →

7 (a) Write down the first ten multiples of 2.
(b) Write down the first ten multiples of 3.
(c) Hence write down the first three multiples common to 2 and 3.
(d) From your answer in (c), what can you say about multiples common to 2 and 3?

8 Determine whether each statement is TRUE or FALSE. Give a counterexample if false.
(a) If 6 is a factor of a number, then 3 is a factor of the number. (b) If 2 and 7 are factors of a number, then 14 is a factor of the number. (c) If 2 and 8 are factors of a number, then 16 is a factor of the number.

填 T（True）或 F（False）。Counterexample: 一个能证明命题不成立的特例。

9 If the product of two whole numbers is 37, what is the sum of these two numbers? Explain your answer.

看完答案，自评：

10 (a) Explain how you can find the number at the top of the tree without finding the other two missing numbers.
(b) Complete the factor tree.

Advanced

11 Identical square tiles are arranged to form rectangular arrays.

Square tiles arranged with a cross-shaped gap — a non-rectangular layout, in contrast to the rectangular arrays the question asks about

Find the number of possible different rectangular arrays if there are
(a) 18 tiles →
(b) 41 tiles →

提示: 数有几对因数（无序），每对对应一种长×宽。

A combination padlock with a 3-digit numeric keypad — the physical lock referenced in the question

A lock can only be opened using a 3-digit number. Jinlan sets this 3-digit number to be the smallest prime number greater than 100. What is this 3-digit number?
Answer:

A green 3D rectangular cuboid — visually cues this is a 3-D volume problem (length × width × height), not a 2-D area problem

The volume of a rectangular box is $40 \text{ cm}^3$. What are two possible sets of dimensions of the box given that they are all whole numbers?

14 What values are possible for the last digit of a 2-digit prime number? Explain your answer.
Possible last digits:

15 A teacher divides 60 students into equal groups. Each group has more than 2 students and fewer than 8 students. How many ways can she group them?
Answer:

提示：找 60 的因数中介于 3 和 7（含）之间的有几个。3, 4, 5, 6 都行；7 不行（60 ÷ 7 不是整数）。

16 Find two consecutive numbers that are both primes. Are there any other consecutive numbers that are both primes? Explain your answer.
Pair:

17 The prime factorisation of a number is $2^4 \times 3^5 \times 7^2 \times 11$. Write down three factors of the number that are greater than 100.

章末概念检查 · Concept Checkpoints

5 道封闭题，自动判分。这些题会进入跨章节复习池——后面学 HCF / LCM 之前会重新弹出，确保你掌握 1.1 的核心。