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# 数学代写|数论代写Number Theory代考|The Greatest Common Divisor

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## 数学代写|数论代写Number Theory代考|The Greatest Common Divisor

It will turn out that the real importance of the Euclidean algorithm is not so much that it can be used to find the greatest common divisor of two numbers, but, as we saw in the example above, it can be used to express that greatest common divisor as a linear combination of the two numbers. In other words, if $a$ and $b$ are two natural numbers, then the Euclidean algorithm will first produce the greatest common divisor, which we write as $\operatorname{gcd}(a, b)$, and then, by reversing the steps in the algorithm, the greatest common divisor can be expressed as a linear combination of the two numbers-that is, it can be written as
$$\operatorname{gcd}(a, b)=x a+y b .$$
Our use of the Euclidean algorithm will often be in the case where the two numbers in question are relatively prime -that is, when their greatest common divisor is 1 -and so, for just a bit more practice, we will now use the Euclidean algorithm not only to show that the two numbers 2001 and 1984 are relatively prime, but to express their greatest common divisor in the form $1=x \cdot 2001+y \cdot 1984$.

The steps in the Euclidean algorithm produce the following sequence of equations:
\begin{aligned} & 2001=1 \cdot 1984+17, \ & 1984=116 \cdot 17+12, \end{aligned}
$$17=1 \cdot 12+5$$

$12=2 \cdot 5+2$,
$5=2 \cdot 2+1$
$2=2 \cdot 1+0$.

## 数学代写|数论代写Number Theory代考|The Division Algorithm

The key step that we repeat over and over again in the Euclidean algorithm, dividing a smaller number into a larger number to get a quotient and a remainder, is itself fundamental enough to warrant a name of its own, the division algorithm:
Given any two integers $a$ and $b$ with $b>0$, there exist unique integers $q$ and $r$ such that $a=q b+r$ with $0 \leq r<b$.
We think of dividing a number $b$ into a number $a$ as many times as we can until the remainder $r$ is smaller than $b$; Euclid, in the Elements, thought of marking off identical line segments of length $b$ along a line segment $a$ until a segment was left whose length $r$ was less than $b$. Either way you think about it, it should be obvious that a unique quotient $q$, and unique remainder $r$ are the result.

Note that care has been taken to express the division algorithm in such a way that negative values of $a$ are possible. So, not only does the division algorithm apply in cases that Euclid might have imagined such as $a=26, b=7$, where we get
$$26=3 \cdot 7+5$$
but it also applies to cases such as $a=-17, b=5$, where we get
$$-17=(-4) \cdot 5+3 .$$

## 数学代写|数论代写Number Theory代考|The Greatest Common Divisor


\operatorname{gcd}(a, b)=x a+y b。




& 2001=1 \cdot 1984+17， \
& 1984=116 \cdot 17+12，



17=1 \cdot 12+5


$12=2 \cdot 5+2$，
$5=2 \cdot 2+1$
$2=2 \cdot 1+0$。

## 数学代写|数论代写Number Theory代考|The Division Algorithm


26=3 \cdot 7+5



-17=(-4) \cdot 5+3。


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