Prime Numbers & Prime Factorisation: The Story of 137 | 质数与质因数分解：137 的故事

📚 Prime Numbers & Prime Factorisation: The Story of 137 | 质数与质因数分解：137 的故事

In this revision guide, we will explore prime numbers and prime factorisation using the number 137 as our special companion. 137 is not only a prime number but also a wonderful example for understanding several key IGCSE Edexcel Mathematical concepts including factor tests, the sieve of Eratosthenes, and finding HCF and LCM.

在这篇复习指南中，我们将以数字 137 作为特殊伙伴，探索质数和质因数分解。137 不仅是一个质数，更是理解 IGCSE Edexcel 数学多个核心概念（如因数检验、埃拉托斯特尼筛法、求最大公因数和最小公倍数）的绝佳例子。

1. What Defines a Prime Number? | 质数的定义

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

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

Important facts to remember:

重要事实需要牢记：

• The number 1 is not prime because it has only one factor.

  数字 1 不是质数，因为它只有一个因数。

• 2 is the smallest and the only even prime number.

  2 是最小且唯一是偶数的质数。

• Every prime number greater than 2 is odd.

  所有大于2的质数都是奇数。

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2. Meet 137: A Prime Number | 认识137：一个质数

Is 137 prime? Let us check its factors. It is odd, so not divisible by 2. The sum of its digits is 1 + 3 + 7 = 11, which is not a multiple of 3, so 137 is not divisible by 3. It does not end in 0 or 5, so it is not divisible by 5. What about 7 and 11?

137 是质数吗？让我们检查它的因数。它是奇数，因此不能被2整除。它的各位数字之和是 1 + 3 + 7 = 11，不是3的倍数，所以137不能被3整除。它不以0或5结尾，所以不能被5整除。那么7和11呢？

7 × 19 = 133, and 7 × 20 = 140, so 7 does not divide 137. Also 11 × 12 = 132 and 11 × 13 = 143, so 11 does not divide 137. Therefore 137 has no prime factors other than itself and 1. We conclude that 137 is a prime number.

7 × 19 = 133，7 × 20 = 140，所以7不能整除137。同样，11 × 12 = 132，11 × 13 = 143，所以11也不能整除137。因此，137除了1和它本身以外没有其他质因数。我们得出结论：137 是一个质数。

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3. How to Test If a Number Is Prime | 如何检验一个数是否为质数

To test whether a number n is prime, you only need to test divisibility by prime numbers up to √n. If none divides n, then n is prime.

要检验一个整数 n 是否为质数，只需要用不超过 √n 的质数去试除。如果这些质数都不能整除 n，那么 n 就是质数。

√137 ≈ 11.7

So we test 137 with the primes 2, 3, 5, 7 and 11. Since none of them divides 137, we are sure 137 is prime.

因此我们用质数 2、3、5、7、11 来检验137。由于它们都不能整除137，我们就可以确定137是质数。

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4. The Sieve of Eratosthenes | 埃拉托斯特尼筛法

One way to find all prime numbers up to a given limit is the Sieve of Eratosthenes. Here is the method:

寻找某个范围内所有质数的一种方法是埃拉托斯特尼筛法。具体步骤如下：

• Write all integers from 2 to the limit in a list.

  把从2到上限的所有整数写在一个列表中。

• Start with the smallest prime 2 and cross out all multiples of 2 after it.

  从最小的质数2开始，划掉2之后所有2的倍数。

• Move to the next uncrossed number, 3, and cross out all multiples of 3 after it.

  移到下一个未被划掉的数3，划掉3之后所有3的倍数。

• Repeat this process. The numbers that remain uncrossed are prime.

  重复这个过程。最后未被划掉的数字就是质数。

Using this sieve up to 137 would verify that 137 remains uncrossed and therefore is prime.

用这个筛法筛到137时，137会一直保持未被划掉，因此它是质数。

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5. Prime Factorisation | 质因数分解

Every integer greater than 1 can be written as a product of prime factors. For example, 84 = 2² × 3 × 7. This process is called prime factorisation.

每一个大于1的整数都可以写成质因数的乘积。例如，84 = 2² × 3 × 7。这个过程称为质因数分解。

Since 137 is prime, its prime factorisation is simply:

由于137是质数，它的质因数分解非常简单：

137 = 137

For a composite number such as 216, we can use a factor tree to get 216 = 2³ × 3³.

对于像216这样的合数，我们可以使用因数分解树得到 216 = 2³ × 3³。

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6. Unique Factorisation Theorem | 唯一分解定理

The Fundamental Theorem of Arithmetic states that every integer greater than 1 has exactly one prime factorisation, apart from the order of the factors. This is why prime factorisation is so reliable in mathematics.

算术基本定理指出：每一个大于1的整数，都有唯一的质因数分解形式，仅除了因数的顺序不同。这就是为什么质因数分解在数学中如此可靠。

For 137, the unique factorisation is simply the prime itself, which shows that prime numbers behave as the “building blocks” of all integers.

对于137来说，它的唯一分解就是质数本身，这表明质数就像所有整数的“积木”。

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7. HCF and LCM Using Prime Factors | 用质因数求最大公因数和最小公倍数

To find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of two numbers, write each number as a product of primes. For HCF, take each common prime with the smaller index. For LCM, take every prime that appears with the larger index.

要求两个数的最大公因数（HCF）和最小公倍数（LCM），先把每个数写成质因数的乘积。对于HCF，取每个公共质因数的最小指数；对于LCM，取每一个出现的质因数的最大指数。

Example with 137 and 84:

以137和84为例：

137 = 137, 84 = 2² × 3 × 7

Because 84 does not contain the prime 137, the HCF is 1. The LCM is 137 × 84 = 11508.

由于84不含质因数137，所以HCF为1。LCM = 137 × 84 = 11508。

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8. Powers and Indices | 幂与指数

Prime factorisation naturally leads to index notation. When a prime is repeated, we write it with a power. For example, 2³ means 2 × 2 × 2 and 137² = 137 × 137 = 18769.

质因数分解自然引入指数记号。当一个质数重复出现时，我们用幂来表示。例如，2³ 表示 2 × 2 × 2，137² = 137 × 137 = 18769。

If you know the prime factorisation of a number, you can use index laws to multiply or divide powers. For example, 137² × 137³ = 137⁵.

如果你知道一个数的质因数分解，就可以利用指数法则进行幂的乘除。例如，137² × 137³ = 137⁵。

137² × 137³ = 137⁵

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9. Why Prime Numbers Matter in Real Life | 质数在现实生活中的重要性

Prime numbers are essential in modern cryptography. Many encryption systems, such as RSA, rely on the fact that multiplying two large primes is easy, but factoring their product is extremely difficult.

质数在现代密码学中至关重要。许多加密系统（如 RSA）依赖于以下事实：将两个大质数相乘很容易，但把它们的乘积分解却极其困难。

Prime numbers are also used in random number generation, hash tables, and internet security protocols. The number 137 itself appears in physics as the fine-structure constant, but in mathematics it remains a beautiful prime.

质数也用于随机数生成、哈希表和互联网安全协议。137 在物理学中还以精细结构常

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