📚 Prime Factors, HCF and LCM | 质因数、最大公因数与最小公倍数
In KS3 Cambridge Mathematics, prime factors, highest common factor (HCF) and lowest common multiple (LCM) are core number skills. They appear in many topics, from simplifying fractions to solving real-life problems. This article explains the key ideas and gives you clear methods to use in class and in exams.
在 KS3 剑桥数学中，质因数、最大公因数（HCF）和最小公倍数（LCM）是核心的数运算技能。它们出现在许多主题中，从化简分数到解决实际问题。本文将解释关键概念，并给出清晰的方法，帮助你在课堂和考试中使用。
1. Factors and Multiples | 因数与倍数
A factor is a whole number that divides exactly into another number with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6 and 12 because each of these divides 12 exactly.
因数是指能够整除另一个数且没有余数的整数。例如，12 的因数是 1、2、3、4、6 和 12，因为这些数都能整除 12。
tiples of 7 are 7, 14, 21, 28 and 35 because they come from 7 × 1, 7 × 2, 7 × 3 and so on.
倍数是一个数与任意整数相乘所得的积。7 的前几个倍数是 7、14、21、28 和 35，因为它们分别来自 7 × 1、7 × 2、7 × 3 等。
- Factors are always less than or equal to the number.
- Multiples are always greater than or equal to the number.
- 因数总是小于或等于这个数。
- 倍数总是大于或等于这个数。
2. Prime Numbers and Composite Numbers | 质数与合数
A prime number is a whole number greater than 1 that has exactly two distinct factors: 1 and itself. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.
质数是大于 1 且恰好只有两个不同因数的整数：1 和它本身。前十个质数是 2、3、5、7、11、13、17、19、23 和 29。
The number 1 is not prime because it has only one factor. Numbers with more than two factors are called composite numbers; for example, 8 has factors 1, 2, 4, 8, so it is composite.
数字 1 不是质数，因为它只有一个因数。拥有两个以上因数的数称为合数；例如 8 的因数是 1、2、4、8，所以它是合数。
Note that 2 is the only even prime number. Every other even number can be divided by 2, so it has at least three factors: 1, 2 and itself.
注意，2 是唯一的偶质数。其他所有偶数都能被 2 整除，因此它们至少有三个因数：1、2 和它本身。
3. What Is Prime Factorisation? | 什么是质因数分解？
Prime factorisation means writing a number as a product of prime numbers only. Every composite number can be broken down into prime factors in exactly one way, if you ignore the order of the factors. This is called the fundamental theorem of arithmetic.
质因数分解是指将一个数写成仅由质数相乘的形式。每个合数都可以用唯一的一种方式分解为质因数（忽略因数的顺序），这称为算术基本定理。
36 = 2 × 2 × 3 × 3 = 2² × 3²
The small raised numbers are called powers or indices, and they tell you how many times each prime factor is repeated.
用指数形式表示时，36 写为 2² × 3²。上标小数字称为幂或指数，它们表示每个质因数被重复乘了多少次。
Prime factorisation is useful because it reveals the building blocks of a number. Once you know these building blocks, you can find HCF and LCM quickly.
质因数分解很有用，因为它揭示了一个数的基本构成。一旦你知道了这些基本构成，就能快速求出 HCF 和 LCM。
4. Using Factor Trees | 使用因数树
A factor tree is a simple diagram that helps you split a number into factors step by step until every branch ends in a prime number. Start by writing any factor pair of the number, then keep splitting composite factors.
因数树是一种简单的图，可以帮助你一步步把一个数分解为因数，直到每条分支的末尾都是质数。先写出这个数的任意一组因数，然后继续拆分合数因数。
plit 10 into 2 and 5. The prime factors at the ends are 2, 2, 3 and 5, so 60 = 2² × 3 × 5.
例如，分解 60 时，先把 60 分成 6 和 10。然后把 6 分成 2 和 3，把 10 分成 2 和 5。末端的质因数是 2、2、3 和 5，因此 60 = 2² × 3 × 5。
nal prime factorisation will be the same. Starting with 60 = 4 × 15 also gives 2² × 3 × 5.
无论你从哪一组因数开始，最终的质因数分解结果都是相同的。从 60 = 4 × 15 开始也能得到 2² × 3 × 5。
5. Highest Common Factor (HCF) | 最大公因数
The highest common factor, or HCF, of two or more numbers is the largest factor that they all share. For 24 and 36, the common factors are 1, 2, 3, 4, 6 and 12, so the HCF is 12.
最大公因数（HCF）是两个或多个数共有的最大因数。对于 24 和 36，公因数是 1、2、3、4、6 和 12，因此 HCF 为 12。
You can list all factors: factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24; factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. The largest number in both lists is 12.
你可以列出所有因数：24 的因数是 1、2、3、4、6、8、12、24；36 的因数是 1、2、3、4、6、9、12、18、36。两个列表中都出现的最大数是 12。
The HCF is sometimes called the greatest common divisor, or GCD. It is always less than or equal to the smallest of the numbers involved.
HCF 有时也被称为最大公约数（GCD）。它总是小于或等于这些数中最小的那个数。
6. Lowest Common Multiple (LCM) | 最小公倍数
The lowest common multiple, or LCM, of two or more numbers is the smallest multiple that they all share. For 4 and 6, the first few multiples are 4, 8, 12, 16, 20, 24 and 6, 12, 18, 24, so the LCM is 12.
最小公倍数（LCM）是两个或多个数共有的最小倍数。对于 4 和 6，前几个倍数分别是 4、8、12、16、20、24 和 6、12、18、24，因此 LCM 是 12。
Listing multiples is fine for small numbers, but for larger numbers it is more efficient to use prime factorisation. This method also helps you avoid missing repeated factors.
对于较小的数，列出倍数的方法是可行的，但对于较大的数，使用质因数分解更高效。这种方法还能帮助你避免遗漏重复的因数。
Remember that the LCM is always greater than or equal to the largest of the numbers involved. If two numbers have no common factor other than 1, their LCM is their product.
请记住，LCM 总是大于或等于这些数中最大的那个数。如果两个数除了 1 之外没有公因数，那么它们的 LCM 就是它们的乘积。
7. HCF and LCM from Prime Factorisation | 用质因数分解求 HCF 与 LCM
Once you have written both numbers as products of primes, you can find the HCF by multiplying every prime factor that appears in both numbers, using the lowest power.
把两个数写成质因数乘积后，你可以通过将两个数中都出现的每个质因数取最低次幂后相乘来求 HCF。
24 = 2³ × 3 and 36 = 2² × 3²
HCF = 2² × 3 = 12
that appears in either number, using the highest power. For 24 and 36, LCM = 2³ × 3² = 72.
对于 LCM，将任意一个数中出现的每个质因数取最高次幂后相乘。对于 24 和 36，LCM = 2³ × 3² = 72。
LCM = 2³ × 3² = 72
This rule is very reliable: lowest powers give HCF, highest powers give LCM. Practise it until it becomes automatic.
这条规则非常可靠：最低次幂得到 HCF，最高次幂得到 LCM。反复练习，直到它变成习惯。
8. Venn Diagrams for HCF and LCM | 用维恩图求 HCF 与 LCM
A Venn diagram is a visual way to compare prime factors. Put the common prime factors in the overlap of two circles, and the remaining prime factors in the separate parts of each circle.
维恩图是比较质因数的可视化方法。把共同的质因数放在两个圆的交叠部分，把剩下的质因数放在每个圆单独的部分。
2, 3, 3. The common part contains 2, 2, 3. The HCF is the product of the overlap: 2 × 2 ×
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