Fact Finders: Mastering Factors, Primes, HCF & LCM | 因数探索：掌握因数、质数、最大公因数与最小公倍数

📚 Fact Finders: Mastering Factors, Primes, HCF & LCM | 因数探索：掌握因数、质数、最大公因数与最小公倍数

Welcome to Fact Finders! In this revision guide, you will learn how to identify factors, break numbers into prime factors, and use them to find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM). This is a core topic in Edexcel IGCSE Mathematics.

欢迎来到“因数探索”！在本复习指南中，你将学习如何识别因数、将数字分解为质因数，并利用它们求最大公因数（HCF）和最小公倍数（LCM）。这是爱德思IGCSE数学的核心考点。

1. What Are Factors? | 什么是因数？

A factor of a number is a whole number that divides exactly into it without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6 and 12.

一个数的因数是指能整除该数且没有余数的整数。例如，12的因数是1、2、3、4、6和12。

Every number has at least two factors: 1 and itself. Factors are usually written in ascending order.

每个数至少有两个因数：1和它本身。因数通常按升序排列。

2. Prime Numbers and Composite Numbers | 质数与合数

A prime number has exactly two distinct positive factors: 1 and itself. For example, 2, 3, 5, 7, 11 and 13 are prime numbers.

质数恰好有两个不同的正因数：1和它本身。例如，2、3、5、7、11和13都是质数。

A composite number has more than two factors. For instance, 6 has factors 1, 2, 3 and 6, so it is composite. The number 1 is neither prime nor composite.

合数有两个以上因数。例如，6的因数是1、2、3和6，因此它是合数。数字1既不是质数也不是合数。

3. Prime Factorisation | 质因数分解

Prime factorisation means expressing a number as a product of its prime factors. For example, 18 = 2 × 3 × 3 = 2 × 3².

质因数分解是指将一个数表示为质因数的乘积。例如，18 = 2 × 3 × 3 = 2 × 3²。

Edexcel IGCSE questions often ask you to write a number in the form of prime factors. Use index notation when a prime factor repeats.

爱德思IGCSE考试常要求你以质因数的形式写出一个数。当质因数重复出现时，使用指数记法。

4. Factor Trees | 因数树

A factor tree is a visual method to break a number into prime factors. Keep dividing until all branches end in prime numbers.

因数树是一种将数字拆分为质因数的直观方法。持续分解，直到所有分支的末端都是质数。

For example, factorising 60: 60 = 6 × 10 = (2 × 3) × (2 × 5) = 2² × 3 × 5.

例如，分解60：60 = 6 × 10 = (2 × 3) × (2 × 5) = 2² × 3 × 5。

60 = 2 × 2 × 3 × 5 = 2² × 3 × 5

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

The Highest Common Factor (HCF) of two or more numbers is the largest number that divides each of them exactly. For example, the HCF of 12 and 18 is 6.

两个或多个数的最大公因数（HCF）是能同时整除它们的最大的数。例如，12和18的最大公因数是6。

To find the HCF by listing factors: list all factors of each number, then pick the largest number that appears in both lists.

通过列举因数求最大公因数：列出每个数的所有因数，然后选择两个列表中都出现的最大的数。

6. Lowest Common Multiple (LCM) | 最小公倍数

The Lowest Common Multiple (LCM) of two or more numbers is the smallest positive number that is a multiple of each of them. For example, the LCM of 4 and 6 is 12.

两个或多个数的最小公倍数（LCM）是能同时被它们整除的最小的正数。例如，4和6的最小公倍数是12。

Multiples of 4: 4, 8, 12, 16, … Multiples of 6: 6, 12, 18, … The first common multiple is 12, so LCM = 12.

4的倍数：4、8、12、16……6的倍数：6、12、18……第一个公倍数是12，因此LCM = 12。

7. HCF and LCM from Prime Factorisation | 用质因数分解求HCF与LCM

Using prime factorisation is more efficient for large numbers. Write each number as a product of primes, then compare the powers.

对于较大的数字，使用质因数分解更高效。将每个数写成质数的乘积，然后比较指数。

For HCF: take the lowest power of every common prime factor. For example, 60 = 2² × 3 × 5 and 72 = 2³ × 3². Common primes: 2 and 3. Lowest powers: 2² and 3¹, so HCF = 2² × 3 = 12.

求HCF：取每个公共质因数的最低指数。例如，60 = 2² × 3 × 5，72 = 2³ × 3²。公共质因数为2和3。最低指数为2²和3¹，因此HCF = 2² × 3 = 12。

For LCM: take the highest power of every prime factor that appears in either number. For 60 and 72, primes are 2, 3 and 5. Highest powers: 2³, 3² and 5¹. LCM = 8 × 9 × 5 = 360.

求LCM：取任一数中出现过的每个质因数的最高指数。对于60和72，质因数为2、3和5。最高指数为2³、3²和5¹。LCM = 8 × 9 × 5 = 360。

8. Real-Life Applications | 实际应用

HCF and LCM appear in real-life problems such as arranging items into rows, timing repeating events, and splitting quantities evenly.

最大公因数和最小公倍数出现在实际问题中，例如将物品排列成行、重复事件的时间计算以及平均分配数量。

Example: Two buses leave the station every 12 minutes and 18 minutes respectively. If they leave together now, after how many minutes will they next leave together? Answer: LCM(12, 18) = 36 minutes.

示例：两辆公交车分别每12分钟和18分钟发车一次。如果它们现在同时发车，再过多少分钟会再次同时发车？答案：LCM(12, 18) = 36分钟。

9. Common Mistakes and Tips | 常见错误与小贴士

Watch out for these common errors:

请注意以下常见错误：

Confusing factors and multiples: factors divide a number, multiples are numbers you get by multiplying a number by integers.

混淆因数与倍数：因数能整除一个数，倍数是一个数乘以整数得到的数。
Forgetting that 1 is not a prime number.

忘记1不是质数。
When using prime factorisation for HCF, choosing the lowest power; for LCM, choose the highest power.

使用质因数分解时，求HCF选最低指数；求LCM选最高指数。

Tip: Always write down your factor tree neatly and check that the product of the prime factors equals the original number.

小贴士：整洁地写出因数树，并检查所有质因数的乘积是否等于原数。

10. Practice Questions | 练习

Try these Edexcel-style questions:

试试这些爱德思风格的题目：

1. Write 84 as a product of prime factors.

1. 将84写成质因数的乘积。

2. Find the HCF and LCM of 24 and 36.

2. 求24和36的最大公因数和最小公倍数。

3. A shop sells packets of pens: one pack has 8 pens, another has 12 pens. What is the least number of pens you can buy so that you have the same number from each pack? (Hint: find LCM)

3. 一家商店出售笔的包装：一种每包8支，另一种每包12支。要使从两种包装买到的笔数相同，最少需要买多少支笔？（提示：求最小公倍数）

Answers: (1) 84 = 2² × 3 × 7. (2) HCF = 12, LCM = 72. (3) LCM(8,12) = 24 pens.

答案：（1）84 = 2² × 3 × 7。（2）HCF = 12，LCM = 72。（3）LCM(8,12) = 24支笔。

11. Summary | 总结

You have learned how to find factors, identify prime numbers, use factor trees, and calculate HCF and LCM using prime factorisation. Keep practising, and these questions will become straightforward.

你已经学会了如何求因数、识别质数、使用因数树，以及利用质因数分解计算HCF和LCM。继续练习，这类题目就会变得简单。

Remember: HCF uses lowest powers of common primes, LCM uses highest powers of all primes.

记住：HCF使用公共质数的最小指数，LCM使用所有质数的最大指数。

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Fact Finders: Mastering Factors, Primes, HCF & LCM | 因数探索：掌握因数、质数、最大公因数与最小公倍数