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By admin 数学代写 2022年2月22日

Definition 2.8.1 The number a divides the number $b$ if, in Theorem 2.7.11. $r=0$. That is, there is zero remainder. The notation for this is $a \mid b$, read a divides $b$ and $a$ is called a factor of $b$. A prime number is one which has the property that the only numbers which divide it are itself and 1 and it is at least 2. The greatest common divisor of two positive integers $m, n$ is that number $p$ which has the property that $p$ divides both $m$ and $n$ and also if $q$ divides both $m$ and $n$, then $q$ divides $p$. Two integers are relatively prime if their greatest common divisor is one. The greatest common divisor of $m$ and $n$ is denoted as $(m, n)$.
There is a phenomenal and amazing theorem which relates the greatest common divisor to the smallest number in a certain set. Suppose $m, n$ are two positive integers. Then if $x, y$ are integers, so is $x m+y n$. Consider all integers which are of this form. Some are positive such as $1 m+1 n$ and some are not. The set $S$ in the following theorem consists of exactly those integers of this form which are positive. Then the greatest common divisor of $m$ and $n$ will be the smallest number in $S$. This is what the following theorem says.

定义 2.8.1 如果在定理 2.7.11 中，数 a 除以数 $b$。 $r=0$。也就是说，余数为零。这个符号是$a \mid b$，读a 除$b$，$a$ 称为$b$ 的因数。素数是一个具有这样的性质的数，即唯一能除它的数是它自己和 1，并且它至少是 2。两个正整数 $m,n$ 的最大公约数是具有这个性质的数 $p$ $p$ 可以同时除以 $m$ 和 $n$，并且如果 $q$ 可以同时除以 $m$ 和 $n$，那么 $q$ 也可以除以 $p$。两个整数互质，如果它们的最大公约数是一。 $m$ 和$n$ 的最大公约数记为$(m, n)$。有一个惊人的定理将最大公约数与某个集合中的最小数联系起来。假设 $m, n$ 是两个正整数。那么如果 $x, y$ 是整数，那么 $x m+y n$ 也是。考虑所有具有这种形式的整数。有些是积极的，例如 $1 m+1 n$，有些则不是。以下定理中的集合 $S$ 恰好由这种形式的正整数组成。那么 $m$ 和 $n$ 的最大公约数将是 $S$ 中的最小数。这就是下面的定理所说的。