Prime Factorisation and Standard Form | 质因数分解与科学计数法

📚 Prime Factorisation and Standard Form | 质因数分解与科学计数法

In this revision article, we explore two essential number topics in the Edexcel IGCSE Maths specification: prime factorisation and standard form. To make the ideas concrete, we will use the number 118 as our running example throughout.

在本复习文章中，我们将探讨爱德思IGCSE数学考纲中两个重要的数论主题：质因数分解与科学计数法。为了使概念更具体，我们将全程使用数字118作为示例。

1. Understanding Factors and Multiples | 理解因数与倍数

A factor of a number is an integer that divides the number exactly, leaving no remainder. For example, the factors of 118 are 1, 2, 59 and 118, because 118 ÷ 1 = 118, 118 ÷ 2 = 59, 118 ÷ 59 = 2 and 118 ÷ 118 = 1.

一个数的因数是能整除该数的整数，且没有余数。例如，118的因数是1、2、59和118，因为118 ÷ 1 = 118，118 ÷ 2 = 59，118 ÷ 59 = 2，118 ÷ 118 = 1。

A multiple of a number is the result of multiplying that number by an integer. The first few multiples of 118 are 118, 236, 354, 472 and 590.

一个数的倍数是将该数与整数相乘的结果。118的前几个倍数是118、236、354、472和590。

• A number may have many factors, but its prime factors are unique.
• Every whole number greater than 1 is either prime or composite.

• 一个数可能有许多因数，但它的质因数是唯一的。
• 每个大于1的整数不是质数，就是合数。

When classifying a number, check whether it is even, whether it ends in 0 or 5, and whether the sum of its digits is a multiple of 3. For 118, the last digit is even, so 2 is a factor.

对任意数字进行分类时，请检查它是否为偶数、末位是否为0或5，以及各位数字之和是否为3的倍数。对于118，末位是偶数，因此2是它的因数。

2. What is a Prime Number? | 什么是质数？

A prime number is a positive integer greater than 1 that has exactly two different factors: 1 and itself. For example, 2, 3, 5, 7, 11 and 13 are all prime numbers.

质数是大于1且恰好只有两个不同因数的正整数：1和它本身。例如，2、3、5、7、11和13都是质数。

The number 1 is not prime because it has only one factor. The number 2 is the only even prime number, because any other even number has 2 as an extra factor.

数字1不是质数，因为它只有一个因数。数字2是唯一的偶数质数，因为其他任何偶数都有额外的因数2。

To test whether a number is prime, try dividing it by all prime numbers up to its square root. For 59, you should test divisibility by 2, 3, 5 and 7. Since none of these divides 59 exactly, 59 is prime.

要判断一个数是否为质数，请尝试用所有不超过其平方根的质数去除它。对于59，你需要检查它是否能被2、3、5和7整除。由于这些数都不能整除59，所以59是质数。

3. Prime Factorisation of 118 | 118的质因数分解

Prime factorisation means writing a number as a product of its prime factors. We can use a factor tree or repeated division. For 118, the first step is to divide by 2, giving 59.

质因数分解是将一个数写成其质因数的乘积。我们可以使用因数分解树或连续除法。对于118，第一步是除以2，得到59。

118 = 2 × 59

Since 59 is prime, the process stops. The prime factorisation of 118 is therefore 2 × 59. Notice that both factors are prime numbers.

由于59是质数，分解过程结束。因此，118的质因数分解是2 × 59。注意，这两个因数都是质数。

We can verify the answer by multiplying 2 and 59: 2 × 59 = 118. This confirms that the prime factorisation is correct.

我们可以通过相乘2和59来验证答案：2 × 59 = 118。这证实了质因数分解是正确的。

In a factor tree, you break down each composite number until all branches end at primes. For 118, the tree has only one branching: 118 splits into 2 and 59, and both are prime, so the tree is complete.

在因数分解树中，你将每个合数不断拆分，直到所有分支末端都是质数。对于118，分解树只有一个分支：118拆分为2和59，两者都是质数，因此分解树完成。

4. Highest Common Factor (HCF) | 最大公因数

The highest common factor of two numbers is the largest positive integer that divides both numbers exactly. To find the HCF using prime factorisation, list the prime factors of each number and multiply the common prime factors together.

两个数的最大公因数是能同时整除这两个数的最大正整数。要使用质因数分解求最大公因数，请列出每个数的质因数，并将共同的质因数相乘。

As an example, let us find the HCF of 118 and 180. We already know that 118 = 2 × 59. Now factorise 180:

作为例子，让我们求118和180的最大公因数。我们已经知道118 = 2 × 59。现在分解180：

180 = 2² × 3² × 5Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

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