📚 Prime Factorisation, HCF and LCM | 质因数分解、最大公因数与最小公倍数
Mastering prime factorisation is a key skill in KS3 mathematics. It unlocks straightforward ways to find the highest common factor (HCF) and lowest common multiple (LCM) of two or more numbers. This article will guide you through factor trees, repeated division, index form, and how to use prime factors to solve HCF and LCM problems with confidence.
掌握质因数分解是 KS3 数学中的一项关键技能。它为我们提供了求两个或多个数的最大公因数 (HCF) 和最小公倍数 (LCM) 的直接方法。本文将引导学生通过因子树、连续除法、指数形式，以及如何利用质因数自信地解决 HCF 和 LCM 问题。
1. Factors and Multiples | 因数与倍数
A factor of a number is a whole number that divides exactly into it, leaving no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6 and 12.
一个数的因数是能整除该数且没有余数的整数。例如，12 的因数有 1、2、3、4、6 和 12。
A multiple of a number is the product of that number and any integer. The first five multiples of 7 are 7, 14, 21, 28 and 35.
一个数的倍数是该数与任意整数的乘积。7 的前五个倍数是 7、14、21、28 和 35。
Understanding factors and multiples is essential before moving on to prime numbers and prime factorisation.
在进入质数和质因数分解之前，理解因数与倍数至关重要。
2. Prime and Composite Numbers | 质数与合数
A prime number has exactly two distinct factors: 1 and the number itself. The first few primes are 2, 3, 5, 7, 11, 13, 17 and 19.
质数恰好有两个不同的因数：1 和它自身。前几个质数是 2、3、5、7、11、13、17 和 19。
A composite number has more than two factors. For instance, 8 is composite because its factors are 1, 2, 4 and 8. Note that 1 is neither prime nor composite.
合数有多于两个因数。例如，8 是合数，因为它的因数是 1、2、4 和 8。注意 1 既不是质数也不是合数。
3. What is Prime Factorisation? | 什么是质因数分解？
Prime factorisation is the process of expressing a composite number as a product of prime numbers. Every whole number greater than 1 can be written as a unique product of primes – this is the Fundamental Theorem of Arithmetic.
质因数分解是将一个合数表示为质数乘积的过程。每个大于 1 的整数都可以写成唯一的质数乘积——这就是算术基本定理。
For example, the prime factorisation of 30 is 2 × 3 × 5, and for 42 it is 2 × 3 × 7.
例如，30 的质因数分解是 2 × 3 × 5，42 的是 2 × 3 × 7。
4. Prime Factorisation Using a Factor Tree | 使用因子树进行质因数分解
A factor tree is a visual method to break a number down step by step. Start by writing the number at the top. Then split it into any two factors. Circle any prime factors. Repeat until every branch ends in a prime number.
因子树是一种逐步分解数字的可视化方法。从顶部写下该数开始，然后将其拆分为任意两个因数。将任何质因数圈出。重复直到每条分支都以质数结束。
Let' 4, and 4 into 2 × 2, so 8 = 2 × 2 × 2. Split 9 into 3 × 3. The circled primes are three 2's and two 3's. So 72 = 2 × 2 × 2 × 3 × 3.
我们来求 72 的质因数。我们可以从 72 = 8 × 9 开始。然后将 8 分解为 2 × 4，4 分解为 2 × 2，所以 8 = 2 × 2 × 2。把 9 分解为 3 × 3。圈出的质数是三个 2 和两个 3。因此 72 = 2 × 2 × 2 × 3 × 3。
5. Prime Factorisation Using Repeated Division | 使用连续除法进行质因数分解
Another reliable method is repeated division by primes. Divide the number by the smallest prime that goes into it evenly, and keep dividing the quotient by the smallest possible prime until the quotient becomes 1.
另一种可靠的方法是连续除以质数。用能整除该数的最小质数去除，并继续用尽可能小的质数除商，直到商变为 1。
2 = 9; divide by 3: 9 ÷ 3 = 3; divide by 3: 3 ÷ 3 = 1. The prime factors are 2, 2, 2, 3, 3 – exactly the same result.
对于 72，除以 2：72 ÷ 2 = 36；除以 2：36 ÷ 2 = 18；除以 2：18 ÷ 2 = 9；除以 3：9 ÷ 3 = 3；除以 3：3 ÷ 3 = 1。质因数为 2、2、2、3、3 —— 结果完全相同。
6. Writing Prime Factors in Index Form | 用指数形式书写质因数
Once you have the list of prime factors, it is neater to write them using powers. The index (or exponent) tells you how many times the prime appears.
一旦有了质因数列表，用幂的形式书写会更加简洁。指数告诉您该质数出现了多少次。
× 2 × 2 × 3 × 3. This can be written as 2³ × 3². Similarly, 100 = 2 × 2 × 5 × 5 = 2² × 5².
对于 72，我们有 2 × 2 × 2 × 3 × 3。可以写成 2³ × 3²。类似地，100 = 2 × 2 × 5 × 5 = 2² × 5²。
Always use index form when working with HCF and LCM – it makes comparison much easier.
在处理 HCF 和 LCM 时，始终使用指数形式——这会让比较变得容易得多。
7. Highest Common Factor (HCF) – Definition | 最大公因数 (HCF) – 定义
The highest common factor of two or more numbers is the largest number that divides exactly into all of them. It is also called the greatest common divisor (GCD).
两个或多个数的最大公因数是能整除所有这些数的最大数。它也被称为最大公约数 (GCD)。
For example, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. The common factors are 1, 2, 3, 4, 6, 12, so the HCF is 12.
例如，24 的因数有 1、2、3、4、6、8、12、24。36 的因数有 1、2、3、4、6、9、12、18、36。公因数为 1、2、3、4、6、12，因此 HCF 是 12。
8. Finding HCF Using Prime Factorisation | 通过质因数分解求 HCF
To find the HCF using prime factors, first write each number as a product of primes in index form. Then, for each common prime base, take the lowest power that appears in any of the numbers. Multiply these together.
要用质因数求 HCF，首先将每个数写成指数形式的质数乘积。然后，对于每个共同的质数基底，取它在任何数中出现的最小指数。将这些乘在一起。
es are 2 and 3. The lowest power of 2 is 2²; for 3 it is 3¹. So HCF = 2² × 3 = 4 × 3 = 12.
求 48 和 180 的 HCF。48 = 2⁴ × 3¹。180 = 2² × 3² × 5¹。共同的质数是 2 和 3。2 的最低指数是 2²；对于 3 是 3¹。所以 HCF = 2² × 3 = 4 × 3 = 12。
9. Lowest Common Multiple (LCM) – Definition | 最小公倍数 (LCM) – 定义
The lowest common multiple of two or more numbers is the smallest number that is a multiple of all of them. It is the first number that appears in the times tables of both original numbers.
两个或多个数的最小公倍数是所有数的倍数中最小的那个。它是两个原数的乘法表中都出现的第一个数。
or 8 we have 8, 16, 24, 32, 40, ... The smallest common multiple is 24, so LCM(6, 8) = 24.
列出几个倍数：6 的倍数有 6、12、18、24、30、36…… 8 的倍数有 8、16、24、32、40…… 最小的公共倍数是 24，因此 LCM(6, 8) = 24。
10. Finding LCM Using Prime Factorisation | 通过质因数分解求 LCM
To find the LCM using prime factors, express each number in index form. Then, for each prime base that appears in any of the numbers, take the highest power. Multiply these together to get the LCM.
要用质因数求 LCM，将每个数表示为指数形式。然后，对于在任何数中出现的每个质数基底，取最高的指数。将它们相乘就得到 LCM。
est power is 2³; for 3, it is 3²; for 5, it is 5¹. So LCM = 2³ × 3² × 5 = 8 × 9 × 5 = 360.
求 24 和 90 的 LCM。24 = 2³ × 3¹。90 = 2¹ × 3² × 5¹。考虑每个质数：对于 2，最高指数是 2³；对于 3，是 3²；对于 5，是 5¹。所以 LCM = 2³ × 3² × 5 = 8 × 9 × 5 = 360。
11. Using the Relationship Between HCF and LCM | 利用 HCF 与 LCM 的关系
a, b) × LCM(a, b) = a × b. This can be used to find one value if you know the other three.
对于任意两个数 a 和 b，有一个有用的恒等式：HCF(a, b) × LCM(a, b) = a × b。如果已知其中三个量，就可以求出第四个。
6. Their LCM is 540 ÷ 6 = 90. Always check that the result is a multiple of both numbers.
例如：两个数的积为 540，HCF 为 6。它们的 LCM 为 540 ÷ 6 = 90。始终检查结果是否为两个数的倍数。
12. Common Errors and Exam Tips | 常见错误与考试技巧
Mistake 1: Forgetting to use prime numbers only in a factor tree. Always circle primes and stop when every branch ends in a prime.
错误 1：在因子树中忘记只使用质数。务必圈出质数，并在每条分支以质数结束时停下。
Mistake 2: Mixing up HCF and LCM methods. HCF takes the smallest exponent for each common prime; LCM takes the largest exponent for every prime present.
错误 2：混淆 HCF 和 LCM 的方法。HCF 对每个公共质数取最小指数；LCM 对出现的每个质数取最大指数。
Mistake 3: Incorrect index form. Remember that 2 × 2 × 2 is 2³, not 2×3.
错误 3：错误的指数形式。记住 2 × 2 × 2 是 2³，而不是 2×3。
Exam tip: Always double-check by seeing if your HCF divides into the original numbers and if your LCM is divisible by both numbers.
考试技巧：始终通过检查 HCF 是否能整除原数以及 LCM 是否能被两个数整除来进行双重验证。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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