Number Skills with 135: Prime Factors, Indices and Standard Form | 用135学习数字技能：质因数、指数与标准形式

📚 Number Skills with 135: Prime Factors, Indices and Standard Form | 用135学习数字技能：质因数、指数与标准形式

In this revision guide, we will work through the core number topics that appear in every Edexcel IGCSE Mathematics paper. To keep things simple and memorable, we will use the number 135 as a running example throughout the article. This helps you see how prime factors, indices, HCF/LCM, standard form and percentages are all connected.

在本篇复习指南中，我们将梳理Edexcel IGCSE数学试卷中必考的核心数论内容。为了让叙述简洁而难忘，我们以数字135作为贯穿全文的例子。这样能帮助你看到质因数、指数、最大公因数、最小公倍数、标准形式与百分比之间的联系。

1. Prime Factorisation of 135 | 1. 135的质因数分解

Prime factorisation means writing a number as a product of its prime factors. A prime number has exactly two factors: 1 and itself. To factorise 135, divide by the smallest prime that works. Since 135 ends in 5, we can start by dividing by 5.

质因数分解是指把一个数写成其质因数的乘积。质数只有两个因数：1和它本身。分解135时，我们先从最小的可用质数开始试除。由于135末尾是5，可以先除以5。

135 ÷ 5 = 27. Then 27 ÷ 3 = 9, 9 ÷ 3 = 3, and 3 ÷ 3 = 1. Collecting the divisors gives 135 = 3 × 3 × 3 × 5. Using index notation, this is written as:

135 ÷ 5 = 27；接着27 ÷ 3 = 9，9 ÷ 3 = 3，3 ÷ 3 = 1。把除数收集起来，得到135 = 3 × 3 × 3 × 5。用指数形式表示就是：

135 = 3³ × 5

Check the answer by expanding: 3³ = 27 and 27 × 5 = 135. Always verify that the product equals the original number.

验证答案：3³ = 27，27 × 5 = 135。务必核对乘积是否等于原数。

2. Index Laws | 2. 指数定律

Index laws, also called laws of indices, help us simplify expressions with powers. The five most important rules are listed here.

指数定律帮助我们化简含有幂的表达式。最重要的五条规则如下。

• When multiplying, add the powers: aᵐ × aⁿ = aᵐ⁺ⁿ

  乘法时指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ

• When dividing, subtract the powers: aᵐ ÷ aⁿ = aᵐ⁻ⁿ

  除法时指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ

• When raising a power to a power, multiply the powers: (aᵐ)ⁿ = aᵐⁿ

  幂的乘方时指数相乘：(aᵐ)ⁿ = aᵐⁿ

• Any non-zero number raised to the power 0 equals 1: a⁰ = 1

  任何非零数的0次幂等于1：a⁰ = 1

• A negative power means reciprocal: a⁻ⁿ = 1/aⁿ

  负指数表示倒数：a⁻ⁿ = 1/aⁿ

For example, 3² × 3³ = 3⁵ = 243, and (3²)³ = 3⁶ = 729. Also, 3⁻¹ = 1/3. Using 135 = 3³ × 5, we can square 135:

例如，3² × 3³ = 3⁵ = 243，(3²)³ = 3⁶ = 729。同时，3⁻¹ = 1/3。利用135 = 3³ × 5，我们可以计算135的平方：

135² = (3³ × 5)² = 3⁶ × 5² = 729 × 25 = 18225

3. HCF and LCM | 3. 最大公因数与最小公倍数

The Highest Common Factor (HCF) is the largest number that divides two or more numbers exactly. The Lowest Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers. Both can be found using prime factorisation.

最大公因数（HCF）是能同时整除掉若干个数的最大数。最小公倍数（LCM）是能同时被若干个数整除的最小数。两者都可以通过质因数分解求出。

Take 135 and 225. We already know 135 = 3³ × 5. Now factorise 225: 225 = 15 × 15 = (3 × 5) × (3 × 5) = 3² × 5².

以135和225为例。我们已经知道135 = 3³ × 5。现在分解225：225 = 15 × 15 = (3 × 5) × (3 × 5) = 3² × 5²。

For the HCF, take the lowest power of each common prime: 3² and 5. Therefore HCF = 3² × 5 = 9 × 5 = 45.

求HCF时，每个公共质数取最低指数：3²和5。因此HCF = 3² × 5 = 9 × 5 = 45。

For the LCM, take the highest power of each prime that appears. Here we need 3³ and 5². Therefore LCM = 3³ × 5² = 27 × 25 = 675.

求LCM时，每个出现的质数取最高指数：需要3³和5²。因此LCM = 3³ × 5² = 27 × 25 = 675。

Check: 135 = 45 × 3 and 225 = 45 × 5, so 45 is a common factor. Also 675 ÷ 135

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