数学代写|MATH2988 Number theory

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这是悉尼大学 数论课程的代写成功案例。

MATH2988简介

This unit of study is an advanced version of MATH2088, sharing the same lectures but with more advanced topics introduced in the tutorials and computer laboratory sessions.

Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. 

Prerequisites 

At the completion of this unit, you should be able to:

• LO1. understand and use the basic terminology of number theory and cryptography
• LO2. carry out simple number-theoretic computations either with a calculator or using MAGMA
• LO3. apply standard number-theoretic algorithms
• LO4. understand and use some classical and number-theoretic cryptosystems
• LO5. apply standard methods to attack some classical cryptosystems
• LO6. understand the theory underlying number-theoretic algorithms and cryptosystems, including the general properties of primes, prime factorisation, modular arithmetic, divisors and multiplicative functions, powers and discrete logarithms.

MATH2988 Number theory HELP（EXAM HELP， ONLINE TUTOR）

Solution:
$267 x+112 y=3$.
First, we find the GCD of 267 and 112 using Euclid’s algorithm (show your work). It is equal to 1 . Next, we find one solution to $267 x+112 y=1$ by going backwards. We find:
$$
267 \cdot(-13)+112 \cdot(31)=1
$$
Therefore, if we multiply throughout by 3 we get:
$$
267 \cdot(-39)+112 \cdot(93)=3 .
$$
By a theorem in class, all the solutions are:
$$
x=-39+112 n, \quad y=93-267 n
$$
for all integers $n$.
$376 x+72 y=18$.
If you calculate the GCD of 376 and 72 you will find out that it is equal to 8 . However, 8 does not divide 18 . Hence, by a theorem in class, this equation does not have solutions in $x, y$ integers.

Solution:

$203 x+119 y=47,48,50$.
These equations do not have solutions because the GCD of 203 and 119 is equal to 7 and 7 does not divide any of 47,48 or 50 .
$203 x+119 y=49$.
First we use Euclid’s algorithm to find a solution of $203 x+119 y=1$, which can be done because the GCD of 203 and 119 is equal to 1 . (Show your work) We find that:
$$
203 \cdot(-7)+119 \cdot(12)=7
$$
Now multiply the equation by 7 to obtain:
$$
203 \cdot(-49)+119 \cdot(84)=49
$$
Therefore, all the solutions are given by:
$$
x=-49+\frac{119}{7} n=-49+17 n, \quad y=84-\frac{203}{7} n=84-29 n .
$$
Of all these, the smallest is $x=2$ and $y=-3$.

Solution:
Let $(a, b)=d$. Then, by Bezout’s identity, there are $r, s$ integers such that
$$
a r+b s=d .
$$
Since $d$ divides $a$ and $b$, we may divide and get:
$$
\frac{a}{d} r+\frac{b}{d} s=1
$$
Therefore, by Bezout’s identity, the GCD of $\frac{a}{d}$ and $\frac{b}{d}$ must divide 1 and it is thus equal to 1.

Solution:
In order to find any rational solutions of this equation, we use the theorem that says that if $\frac{m}{n}$ is a root, then $m$ is a divisor of 81 and $n$ is a divisor of 5. Thus, $m \in{ \pm 1, \pm 3, \pm 9, \pm 27, \pm 81}$, and $n \in{1,5}$. After checking these possibilities, we find that $x=3 \in \mathbb{N}, x=-9 \in \mathbb{Z}$ and $x=\frac{3}{5}$ are all the roots of the equation.

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