IGCSE Edexcel Mathematics: Prime Factorisation of 203 | IGCSE数学：203的质因数分解

📚 IGCSE Edexcel Mathematics: Prime Factorisation of 203 | IGCSE数学：203的质因数分解

Welcome to this IGCSE Edexcel Mathematics revision session. In this article we will use the number 203 as a focused case study to revise prime factorisation, divisibility, HCF/LCM and surds. These are core topics in the Number syllabus, and mastering them will help you solve both straightforward and multi-step examination questions.

欢迎来到本次IGCSE Edexcel数学复习课。在这篇文章中，我们将以数字203为核心案例，复习质因数分解、整除性、最大公因数/最小公倍数以及根式的化简。这些都是数与代数部分的核心考点，掌握它们有助于你解决直接计算和多步骤的应用题。

1. What is 203? | 203是什么？

203 is a positive integer, located between 202 and 204. It is an odd number because its last digit is 3. Most importantly, 203 is a composite number, not a prime number, because it has positive divisors other than 1 and itself. In the Edexcel IGCSE syllabus, you are expected to classify numbers as prime, composite or neither, and to justify your answer.

203是一个正整数，位于202和204之间。它是一个奇数，因为它的末位数字是3。最重要的是，203是一个合数，而不是质数，因为它除了1和自身以外还有其他正因数。在Edexcel IGCSE考试大纲中，要求你能够判断一个数是质数、合数还是既不质又不合，并说明理由。

A prime number must have exactly two distinct positive factors: 1 and itself. Since we will soon see that 203 can be written as the product of two smaller whole numbers, it automatically fails the definition of a prime number.

质数必须恰好有两个不同的正因数：1和它本身。因为我们将很快看到203可以写成两个更小的整数的乘积，所以它自然不符合质数的定义。

2. Prime Factorisation of 203 | 203的质因数分解

Prime factorisation means expressing a number as a product of prime numbers. To factorise 203, we first test the smallest prime numbers in order: 2, 3, 5, 7, 11, 13, and so on. We only need to test primes up to the square root of 203, because if a factor exists, one factor must be less than or equal to the square root.

质因数分解是指将一个数表示成质数的乘积。要对203进行分解，我们首先按顺序测试最小的质数：2、3、5、7、11、13等。我们只需要测试到203的平方根以内的质数，因为如果存在一个因数，那么其中一个因数必定小于或等于平方根。

Let us test: 203 is not divisible by 2, as it is odd. The digit sum is 2 + 0 + 3 = 5, so it is not divisible by 3. It does not end in 0 or 5, so it is not divisible by 5. When we try 7, we find that 7 × 29 = 203. Since 29 is also a prime number, the factorisation is complete.

我们来测试：203不能被2整除，因为它是奇数。各位数字和为2 + 0 + 3 = 5，所以不能被3整除。它不以0或5结尾，所以不能被5整除。当我们尝试7时，发现7 × 29 = 203。由于29也是质数，分解完成。

203 = 7 × 29

Therefore the prime factorisation of 203 is the product 7 × 29. Because both factors are prime, we cannot factorise any further.

因此，203的质因数分解就是乘积7 × 29。由于两个因数都是质数，我们无法再继续分解。

3. Why is 203 Not a Prime Number? | 为什么203不是质数？

To prove that 203 is not prime, we can use trial division. First estimate the square root: √203 ≈ 14.25. So we only need to check prime numbers less than or equal to 14: 2, 3, 5, 7, 11 and 13.

为了证明203不是质数，我们可以使用试除法。首先估算平方根：√203 ≈ 14.25。因此我们只需要检查不超过14的质数：2、3、5、7、11和13。

We have already shown that 2, 3 and 5 do not divide 203. Next, 7 divides 203 because 7 × 29 = 203. As soon as we find at least one other divisor besides 1 and 203, we can conclude that 203 is composite. In an exam, you would not need to check all primes once you find a factor, but listing the checks shows clear reasoning.

我们已经说明2、3和5都不能整除203。接下来，7能整除203，因为7 × 29 = 203。一旦我们找到除1和203之外的另一个因数，就可以断定203是合数。在考试中，一旦找到一个因数，就不需要再检查所有质数，但列出检查过程可以展示清晰的推理。

√203 ≈ 14.25

The other candidates 11 and 13 do not divide 203 (11 × 18 = 198, 11 × 19 = 209; 13 × 15 = 195, 13 × 16 = 208). This confirms that 7 is the only prime factor less than √203 apart from the complementary factor 29.

其他候选质数11和13都不能整除203（11 × 18 = 198，11 × 19 = 209；13 × 15 = 195，13 × 16 = 208）。这确认了7是小于√203的唯一质因数，而相应的另一个质因数是29。

4. Finding All Factors of 203 | 寻找203的所有因数

Once we have the prime factorisation 203 = 7 × 29, finding all positive factors is straightforward. Each factor must be a product of a selection of the prime factors. The possible factor pairs are (1, 203) and (7, 29).

一旦我们得到质因数分解203 = 7 × 29，找出所有正因数就很简单了。每个因数都必须是这些质因数中某些的乘积。可能的因数对是(1, 203)和(7, 29)。

Thus the complete list of positive factors of 203 is:

因此，203的所有正因数列表为：

1, 7, 29, 203

We can also consider negative factors. Since a negative times a negative gives a positive, the negative factors are -1, -7, -29 and -203. However, IGCSE questions usually ask for positive factors unless stated otherwise.

我们也可以考虑负因数。因为负负得正，所以负因数为-1、-7、-29和-203。然而，IGCSE题目通常只要求正因数，除非另有说明。

Remember that the number 1 and the number itself are always factors of any

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）