什么是质数？定义、类型、示例问题

用于表示和处理数字的方法称为数字系统。数字系统是一种表示数字的书写系统。它是用于通过使用数字或其他符号来表示给定集合的数字的数学符号。它允许进行算术运算，例如除法、乘法、加法、减法。一些重要的数字系统是十进制数字系统，二进制数字系统，八进制数字系统，十六进制数字系统。

十进制数系统

十进制数字系统由十位数字组成，即从0到9。十进制数字系统的基数是10。这些数字可以用来表示或表达任何数值。例如，十进制数 153 由一位数字 3、十位数字 5 和百位数字 1 组成，可以表示为：

(1 × 10 ² ) + (5 × 10 ¹ ) + (3 × 10 ⁰ )

= (1 × 100) + (5 × 10) + (3 × 1)

{其中，100 = 1}

= 100 + 50 + 3

= 153

十进制数系统根据不同的特点有不同的类型，比如有整数、自然数、素数、合数等。下面我们来详细了解一下素数，

什么是质数？

在数字系统中，质数是那些只有两个因子的数字，即 1 和数字本身。换句话说，素数是可以被 1 和数字本身整除的数字。

• 一个素数应该正好包含两个因子。
• 素数应该能被 1 和数字本身整除。

假设 p 是一个素数，那么 p 只有 2 个因子，即 1 和 p 本身。任何不跟随它的数字都称为合数。例如 8 的因子是 1、2、4 和 8，它们总共是四个因子。但是5的因数是1和5本身，完全是两个因数。因此，5 是素数，但 8 不是素数，而是合数。

前十个自然素数是 – 2, 3, 5, 7, 11, 13, 17, 19, 23, 29  

1. Factors of 1 are =1  ( Not Prime Number because it has only one factor)
2. Factors of 2 are = 1 and 2 ( Prime Number because it has only two factors )
3. Factors of 3 are =1 and 3 ( Prime Number because it has only two factors )
4. Factors of 4 are =1, 2, and 4 ( Not Prime Number because it has three factors )
5. Factors of 5 are =1 and 5 ( Prime Number because it has only two factors )
6. Factors of 6 are =1, 2, 3, and 6 ( Not Prime Number because it has four factors )
7. Factors of 7 are =1 and 7 ( Prime Number because it has only two factors )
8. Factors of 8 are =1, 2, 4, and 8 ( Not Prime Number because it has four factors )
9. Factors of 9 are =1, 3, and 9 ( Not Prime Number because it has three factors )
10. Factors of 10 are =1, 2, 5, and 10 ( Not Prime Number because it has four factors )

注意： 1 是非质数，因为根据定义，质数应该只包含两个因数，而 1 只有一个因数。所以1不是素数。

1到100之间的质数列表

众所周知，素数是只有两个因子的数字，即1和数字本身。以上是存在于 1 到 100 之间的质数。

偶数素数

偶数 数字是可以被2整除的数字。因此，2是数字系统中唯一为偶数的素数，因此2被称为偶素数，其余素数是奇数，因此它们被称为奇数素数。

• 偶数素数 = 2
• 奇数素数 = 3、5、7、11、13、17、19、23、29、31、37、41、43、47 等。

双质数

双胞胎 素数是它们之间只有一个合数的数字，称为孪生素数或孪生素数。孪生素数的其他词是两个素数之差仅为2的素数对。几个例子是，

• (5, 7) = 7 – 5 = 2
• (11, 13) = 13 – 11 = 2
• (17, 19) = 19 – 17 = 2
• (41, 43) = 43 – 41 = 2

示例问题

问题1：51是质数吗？

回答 ：

No, because the only factors 51 are 1, 3, 17, and 51. According to the definition, a prime number should contain only two factors. Factors of 51 = 1, 3, 17, 51 ( 4 Factors ). So, 51 is not a prime number.

问题2：1是质数吗？

回答 ：

1 is not a prime number because according to the definition, a prime number should exactly two factors But, number 1 has one and only one factor which is 1 itself. Thus, 1 is not considered a Prime number. Factors of 1 = 1 ( 1 Factor). So, 1 is not a prime number 

问题3：11是质数吗？

回答：

Yes, 11 is a prime number because it is only divisible by 2 numbers that is 1 and the number itself (11). Its has two factors 1 and 11 only. Factors of 11 = 1, 11 ( 2 Factors ). So, 11 is a prime number.

问题 4：找出下列数字 1, 22, 3, 51, 75, 88, 65, 63, 19, 7, 39, 47, 60, 100, 12, 10, 5 中的所有质数？

回答：

All the prime number from the given numbers are – 3, 19, 7, 39, 47, 5

Explanation,

• Factor of 1 are = 1
• Factor of 22 are = 1, 2, 11, 22
• Factor of 3 are = 1, 3  ( only two factors )
• Factor of 51 are = 1, 3, 17, 51
• Factor of 75 are = 1, 3, 5, 15, 25, 75.
• Factor of 88 are = 1, 2, 4, 8, 11, 22, 44, 88
• Factor of 65 are = 1, 5, 65
• Factor of 63 are = 1, 3, 7, 9 , 21, 63
• Factor of 19 are = 1, 19  ( only two factors )
• Factor of 7 are = 1, 7  ( only two factors )
• Factor of 39 are = 1, 39  ( only two factors )
• Factor of 47 are = 1, 47  ( only two factors )
• Factor of 60 are = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
• Factor of 100 are = 1, 2, 4, 5,10, 20, 25, 50, 100
• Factor of 12 are = 1, 2, 3, 4, 6, 12
• Factor of 10 are = 1, 2, 5, 10
• Factor of 5 are = 1, 5  ( only two factors )

问题5：数系中最小的素数是哪个？

回答：

2 is the smallest prime number in the number system. It has only two factors that is 1 and 2. Factors of 2 are : 1, 2

问题6：最大的两位素数是多少？

答案：

97 is the largest two digit prime numbers. Factors of 97 are = 1, 97