📚 Factors, Multiples and Prime Numbers | 因数、倍数与质数
In KS3 mathematics, factors, multiples and prime numbers form the building blocks of number work. They help you simplify fractions, find common denominators and solve a wide range of problems. This article covers the key ideas you need for the Cambridge KS3 curriculum.
在 KS3 数学中，因数、倍数和质数是数字运算的基础。它们帮助你化简分数、寻找公分母并解决各种问题。本文涵盖剑桥 KS3 课程所需的关键概念。
1. What Are Factors? | 什么是因数？
20 because 20 ÷ 5 = 4 exactly. Every whole number has at least two factors: 1 and itself.
一个整数的因数是能整除它的任何整数，并且不留下余数。例如，5 是 20 的因数，因为 20 ÷ 5 = 4 恰好整除。每个整数至少有两个因数：1 和它本身。
To check whether a number is a factor, simply divide. If the answer is an integer, then the number is a factor. For instance, 7 is not a factor of 20 because 20 ÷ 7 gives 2 remainder 6.
要检查一个数是否是因数，只需做除法。如果答案是整数，那么这个数就是因数。例如，7 不是 20 的因数，因为 20 ÷ 7 得 2 余 6。
2. Finding Factor Pairs | 寻找因数对
Factors always occur in pairs that multiply together to give the original number. To find all factor pairs of a number, start with 1 and the number itself, then test 2, 3, 4 and so on until the pairs repeat.
因数总是成对出现，相乘得到原数。要找出一个数的所有因数对，从 1 和该数本身开始，然后依次测试 2、3、4 等，直到因数对重复。
24 = 1 × 24 = 2 × 12 = 3 × 8 = 4 × 6
The factors of 24 are therefore 1, 2, 3, 4, 6, 8, 12 and 24. Notice that 5 is missing because 24 ÷ 5 is not exact.
因此，24 的因数是 1、2、3、4、6、8、12 和 24。注意没有 5，因为 24 ÷ 5 不能整除。
3. Multiples and Common Multiples | 倍数与公倍数
A multiple of a number is the product of that number and any whole number. The first five multiples of 6 are 6, 12, 18, 24 and 30. Unlike factors, every number has infinitely many multiples.
一个数的倍数是该数与任意整数的乘积。6 的前五个倍数是 6、12、18、24 和 30。与因数不同，每个数都有无限多个倍数。
Common multiples are multiples shared by two or more numbers. For example, the common multiples of 4 and 6 include 12, 24, 36 and so on. The smallest of these is called the lowest common multiple.
公倍数是两个或多个数共有的倍数。例如，4 和 6 的公倍数包括 12、24、36 等。其中最小的称为最小公倍数。
- Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36 ...
- Multiples of 6: 6, 12, 18, 24, 30, 36 ...
4. Prime Numbers and Composite Numbers | 质数与合数
A prime number has exactly two distinct factors: 1 and itself. A composite number has more than two factors. The number 1 is special because it has only one factor, so it is neither prime nor composite.
质数恰好有两个不同的因数：1 和它本身。合数有两个以上的因数。数字 1 很特殊，因为它只有一个因数，所以既不是质数也不是合数。
The first eight prime numbers are 2, 3, 5, 7, 11, 13, 17 and 19. Notice that 2 is the only even prime number; every other even number is divisible by 2 and therefore composite.
前八个质数是 2、3、5、7、11、13、17 和 19。注意 2 是唯一的偶数质数；其他所有偶数都能被 2 整除，因此是合数。
5. The Sieve of Eratosthenes | 埃拉托色尼筛法
The Sieve of Eratosthenes is a systematic way to find all prime numbers up to a given limit. You start with a number grid, circle 2 and cross out all multiples of 2, then circle 3 and cross out all multiples of 3, and continue with the next uncrossed number.
埃拉托色尼筛法是一种系统寻找某个范围内所有质数的方法。从一个数字网格开始，圈出 2 并划去 2 的所有倍数，然后圈出 3 并划去 3 的所有倍数，继续处理下一个未划去的数。
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