## 数学代写|数论代写number theory代考MAT115A|The Euclidean algorithm

2022年6月29日

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## 数学代写|数论代写number theory代考|Finding the gcd

Most readers will know the Euclidean algorithm, used to find the greatest common divisor (ged) of two given integers. For example, to determine the greatest common divisor of 85 and 48 , we begin by subtracting the smaller from the larger, 48 from 85 , to obtain $85-48=37$. Now $\operatorname{gcd}(85,48)=\operatorname{gcd}(48,37)$, because the common divisors of 48 and 37 are precisely the same as those of 85 and 48 , and so we apply the algorithm again to the pair 48 and 37. So we subtract the smaller from the larger to obtain $48-37=11$, so that $\operatorname{ged}(48,37)=\operatorname{gcd}(37,11)$. Next we should subtract 11 from 37 , but then we would only do so again, and a third time, so let’s do all that in one go and take $37-3 \times 11=4$, to obtain $\operatorname{gcd}(37,11)=\operatorname{ged}(11,4)$. Similarly we take $11-2 \times 4=3$, and then $4-3=1$, so that the ged of 85 and 48 is 1. This is the Euclidean algorithm that you might already have seen, 1 but did you ever prove that it really works?

To do so, we will first carefully define terms that we have implicitly used in the above paragraph, perhaps mathematical terms that you have used for years (such as “divides”, “quotient”, and “remainder”) without a formal definition. This may seem pedantic but the goal is to make sure that the rules of basic arithmetic are really established on a sound footing.

Let $a$ and $b$ be given integers. We say that $a$ is divisible by $b$, or that $b$ divides $a, 2$ if there exists an integer $q$ such that $a=q b$. For convenience we write ” $b \mid a$ ” $3 .$ We now set an exercise for the reader to check that the definition allows one to manipulate the notion of division in several familiar ways.

## 数学代写|数论代写number theory代考|Linear combinations

The Euclidean algorithm can also be used to determine a linear combination $]^{7}$ of $a$ and $b$, over the integers, which equals $\operatorname{gcd}(a, b)$; that is, one can always use the Euclidean algorithm to find integers $u$ and $v$ such that
$$a u+b v=\operatorname{gcd}(a, b) \text {. }$$
Let us see how to do this in an example, by finding integers $u$ and $v$ such that $85 u+48 v=1$; remember that we found the ged of 85 and 48 at the beginning of section 1.1. We retrace the steps of the Euclidean algorithm, but in reverse: The final step was that $1=1 \cdot 4-1 \cdot 3$, a linear combination of 4 and 3 . The second to last step used that $3=11-2 \cdot 4$, and so substituting $11-2 \cdot 4$ for 3 in $1=1 \cdot 4-1 \cdot 3$, we obtain
$$1=1 \cdot 4-1 \cdot 3=1 \cdot 4-1 \cdot(11-2 \cdot 4)=3 \cdot 4-1 \cdot 11,$$
a linear combination of 11 and 4. This then implies, since we had $4=37-3 \cdot 11$, that
$$1=3 \cdot(37-3 \cdot 11)-1 \cdot 11=3 \cdot 37-10 \cdot 11,$$
a linear combination of 37 and 11. Continuing in this way, we successively deduce, using that $11=48-37$ and then that $37=85-48$,

## 数学代写|数论代写number theory代考|The set of linear combinations of two integers

Theorem 1.1 states that the greatest common divisor of two integers is a linear combination of those two integers. This suggests that it might be useful to study the set of linear combinations
$$I(a, b):={a m+b m: m, n \in \mathbb{Z}}$$
of two given integers $a$ and $b \otimes$ We see that $I(a, b)$ contains $0, a, b, a+b, a+$ $2 b, 2 b+a, a-b, b-a, \ldots$ and any sum of integer multiples of $a$ and $b$, so that $I(a, b)$ is closed under addition. Let $I(a):=I(a, 0)={a m: m \in \mathbb{Z}}$, the set of integer multiples of $a$. We now prove that $I(a, b)$ can be described as the set of integer multiples of ged $(a, b)$, a set which is easier to understand:
Corollary 1.3.1. For any given non-zero integers $a$ and b, we have
$${a m+b n: m, n \in \mathbb{Z}}={g k: k \in \mathbb{Z}}$$
where $g:=\operatorname{gcd}(a, b)$; that is, $I(a, b)=I(g)$. In other words, there exist integers $m$ and $n$ with $a m+b n=c$ if and only if $\operatorname{ged}(a, b)$ divides $c$.

Proof. By Theorem $1.1$ we know that there exist $u, v \in \mathbb{Z}$ for which $a u+b v=g$. Therefore $a(u k)+b(v k)=g k$ so that $g k \in I(a, b)$ for all $k \in \mathbb{Z}$; that is, $I(g) \subset I(a, b)$. On the other hand, as $g$ divides both $a$ and $b$, there exist integers $A, B$ such that $a=g A, b=g B$, and so any $a m+b n=g(A m+B n) \in I(g)$. That is, $I(a, b) \subset I(g)$. The result now follows from the two inclusions.

## 数学代写|数论代写number theory代考|Linear combinations

$$a u+b v=\operatorname{gcd}(a, b)$$

$$1=1 \cdot 4-1 \cdot 3=1 \cdot 4-1 \cdot(11-2 \cdot 4)=3 \cdot 4-1 \cdot 11$$
11 和 4 的线性组合。这意味着，因为我们有 $4=37-3 \cdot 11$ ，那
$$1=3 \cdot(37-3 \cdot 11)-1 \cdot 11=3 \cdot 37-10 \cdot 11$$
37 和 11 的线性组合。以这种方式继续，我们依次推导，使用 $11=48-37$ 然后那个 $37=85-48$,

## 数学代写|数论代写number theory代考|The set of linear combinations of two integers

$$I(a, b):=a m+b m: m, n \in \mathbb{Z}$$

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