🏠 首页 CHAPTER 1 · FACTORS AND MULTIPLES 1.X 章末

1.X

Chapter 1 — Let's Sum Up · Review · Problem Solving · Journal

🎓 第 1 章的章末整套——做完 1.1 / 1.2 / 1.3 / 1.4 后回来：先看 Let's Sum Up 复习核心规则；再做 20 道 Review Exercise 1（综合 prime / HCF / LCM / 平方根立方根）；挑战 Problem Solving Task（100 个储物柜问题）；最后用 Maths Journal 把学到的写下来。

🎯 LET'S SUM UP!

Classification of Whole Numbers

0, 1, 2, 3, 4, … are whole numbers. We classify whole numbers based on the number of factors they have.

  1. A whole number greater than 1 can be classified either as a prime number or a composite number.
  2. A prime number has only two factors, 1 and itself. (e.g., 2, 3, 5, 7, …)
  3. A composite number has more than two factors. (e.g., 4, 6, 8, 9, …)

Prime Factorisation

Prime factorisation is the process of expressing a composite number as a product of prime factors only.

Example: 680.
Method 1: Factor Tree
680 → 10 × 68
10 → 2 × 5; 68 → 2 × 34; 34 → 2 × 17
680 = 2³ × 5 × 17
Method 2: Successive Short Division
2 | 680
2 | 340
2 | 170
5 |   85
17 |   17
    1
680 = 2³ × 5 × 17

Highest Common Factor (HCF) and Lowest Common Multiple (LCM)

We can find the HCF and LCM of two or more numbers using prime factorisation.

HCF

HCF of 700 and 840:

700 = 2² × 5² × 7
840 = 2³ × 3 × 5 × 7

HCF = 2² × 5 × 7 = 140
(multiply common prime factors with the lower power)

LCM

LCM of 700 and 840:

700 = 2² × 5² × 7
840 = 2³ × 3 × 5 × 7

LCM = 2³ × 3 × 5² × 7 = 4200
(multiply all prime factors with the higher power)

Square and Square Root

Square of 3 = 3² = 9.   Positive square root of 9 = √9 = √(3²) = 3.

A number whose square root is a whole number is called a perfect square. (e.g., 1, 4, 9, 16, 25, …)

If the prime factorisation of a number is 2⁶ × 3⁴ × 5¹², is the number a perfect square?

💡 思考后展开 Yes—all exponents are even (6, 4, 12 all even) → it is a perfect square.

Cube and Cube Root

Cube of 5 = 5³ = 125.   Cube root of 125 = ∛125 = ∛(5³) = 5.

A number whose cube root is a whole number is called a perfect cube. (e.g., 1, 8, 27, 64, 125, …)

If the prime factorisation of a number is 7¹² × 13⁹ × 19³, is the number a perfect cube?

💡 思考后展开 Yes—all exponents are multiples of 3 (12, 9, 3) → it is a perfect cube.

🔁 REVIEW EXERCISE 1

本章 20 道综合复习题——结合 1.1（primes）、1.2（HCF）、1.3（LCM）、1.4（平方根/立方根）的所有规则。

  1. Find the smallest number that has 2, 5 and 7 as its prime factors.

    =  

  2. Determine whether each number is prime or composite.

    (a) 649 →  (b) 721 →  

    💡 因式分解

    649 = 11 × 59 → composite。721 = 7 × 103 → composite。

  3. True or False?

    (a) If 3 and 5 are factors of a number, then 15 is a factor.    

    (b) If 246 is a multiple of a number, then 123 is a multiple of the number.    

    💡 解释

    (a) 因 gcd(3,5)=1，所以 3|n 且 5|n → 15|n ✓ TRUE。
    (b) 反例：n=2，246=2×123 是 2 的倍数，但 123 不是 2 的倍数 → FALSE。

  4. Complete the two factor trees:

    (i) 150 (children: 25 and ?, then 25 → ? × ?, ? → ? × ?). Find the result.

    (ii) Top = ? (children: 3 and ?, then ? → 5 × 4, then 4 → 2 × 2).

    (b) Top of tree (i) in index notation: 150 =

    Top of tree (ii) =  in index notation:  

    (c) HCF of 150 and 60 =  (d) LCM =  

  5. Given 12, 40 and 45:

    (a) HCF =  (b) LCM =  

    (c) Greatest 4-digit common multiple =  

  6. Express in prime factorisation (index notation):

    (a)(i) 12 =  (ii) 144 =  (iii) 5040 =

    (b) HCF=12, LCM=5040; one number is 144. Find the other:    

  7. Lollies: 240 per box; Cookies: 75 per pack. Mrs Tan buys the same number of lollies and cookies.

    (a) 240 =

    (b) HCF(75, 240) =

    (c) Least number of boxes of lollies she could buy =  

    💡 推导

    同数 = LCM(240,75) = 1200。1200/240 = 5 boxes; 1200/75 = 16 packs。最少 5 盒糖。

  8. Product of two numbers = 3388, HCF = 11. Find all possible pairs.

    自评：

  9. Rectangle dimensions (2⁵ × 7) cm by (2 × 5² × 7³) cm.

    (a) Area in prime factorisation =

    (b) Square has same area. Side length = cm  

  10. (a) 375 in prime factorisation =

    (b) A square has area (375 × n) cm² where n is a whole number. Smallest integer n =  

  11. (a)(i) √2601 =  (ii) ∛(375 × 243) =

    (b) HCF of the two numbers in (a) =  

  12. The product of three consecutive numbers is 1716. Find the sum of the three numbers.

    =  

    💡 提示

    1716 = 2² × 3 × 11 × 13。试 11×12×13 = 1716 ✓。和 = 11+12+13 = 36。

  13. A number has exactly 12 factors. Three of them are 3, 4, 5. What are the remaining 9?

    自评：

  14. What is the smallest 4-digit number that can be divided by all numbers from 1 to 10 exactly?

    =  

    💡 LCM(1..10)

    LCM(1..10) = 2520（恰好 4 位数）→ 最小 4 位数 = 2520。

  15. Singapore Flyer: total capacity 784 passengers. Each capsule carries the same number = total number of capsules. (a) 784 in prime factorisation. (b) Find the number of capsules.

    (a) 784 =

    (b) Capsules = √784 =  

  16. Ahmad, Bryan, Clara visit a cafe every 6, 8, 15 days. All three visit on 1 January 2019. Next date they all visit together?

    =  

    💡 推导

    LCM(6, 8, 15) = 120 days。1 Jan + 120 days = 1 May 2019。（30+28+31+30+1 = 120 → 5 月 1 日）

  17. Chocolate bars: 18 almonds, 24 hazelnuts, 30 peanuts. Distributed equally among students; each student gets only one type; greatest possible number per student.

    (a) Each student gets bars  (b) Peanut students =  

  18. Bell 1 every 25 min, Bell 2 every 30 min. Bell 1 starts at 5:50 a.m., Bell 2 at 6:00 a.m.

    (a) Next ring together =  

    (b) Times before 6 p.m. they ring together =  

    💡 推导

    找首次同响：试 j=3 时 Bell 2 在 6:00 + 90 = 7:30；Bell 1 在 5:50 + 100 = 7:30 ✓。然后每 LCM(25,30) = 150 min = 2.5 h 同响一次：7:30, 10:00, 12:30, 15:00, 17:30 → 5 次。

  19. Rectangular board 315 cm × 99 cm, divided into squares.

    (a) Largest possible side = cm  (b) 2nd largest = cm  

    (c) Other length(s)?

    自评：

  20. Fruits divided among students:

    • Among 4 students → 2 left over
    • Among 6 students → 4 left over
    • Among 15 students → 13 left over

    Given total ≈ 120, find exact number.

    Total =  

    💡 推导

    2 = 4−2, 4 = 6−2, 13 = 15−2 → total ≡ −2 mod LCM(4,6,15) = 60。total = 60k − 2。约 120 → k=2，total = 118。验证：118÷4=29r2 ✓；118÷6=19r4 ✓；118÷15=7r13 ✓。

🧠 PROBLEM SOLVING TASK — Which lockers are open?

There are 100 closed lockers numbered 1 to 100, and 100 students.

  • The 1st student opens all 100 lockers.
  • The 2nd student changes the status (closes open / opens closed) of every locker that is a multiple of 2.
  • The 3rd student changes the status of every locker that is a multiple of 3.
  • … The k-th student changes the status of every locker that is a multiple of k.

Question: After all 100 students have done their work, which lockers are open?

自评：

额外挑战：1000 个学生 + 1000 个储物柜，结果还成立吗？

💡 思考后展开

仍然成立——开着的柜子是 1 到 1000 之间的完全平方数：1, 4, 9, ..., 961（31²），共 31 个柜子。

📖 MATHS JOURNAL

把学到的写下来——用自己的话整理知识。

1 Write down the steps involved in using prime factorisation to find the HCF and LCM of two numbers. You may use examples to help explain.
自评：
2 Give one example each of how HCF and LCM are applied in real life.
自评：
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