Elementary Number Theory proof

In summary: Therefore, if $k$ divides both $a$ and $b$, it also divides $as+bt$. In summary, to prove that if $k$ divides the integers $a$ and $b$, then $k$ divides $as+bt$ for every pair of integers $s$ and $t$, we can show that $a=km$ and $b=kn$ for some integers $m$ and $n$, and then use this to rewrite $as+bt$ as $k(ms+nt)$, showing that $k$ divides $as+bt$.
  • #1
cbarker1
Gold Member
MHB
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Dear Everyone,

Here is the question:

"Prove that if $k$ divides the integers $a$ and $b$, then $k$ divides $as+bt$ for every pair of integers $s$ and $t$ for every pair of integers."

The attempted work:

Suppose $k$ divides $a$ and $k$ divides $b$, where $a,b\in\mathbb{Z}$. Then, $a=kt$ and $b=ks$, where $s,t\in\mathbb{Z}$ (Here is where I am stuck).

Do I solve for $k$?

If I do solve for $k$, then it yields the
$k=\frac{a}{t}$. Then, $b=\frac{as}{t}$. So $bt-as=0$. Then $0$ divides $bt-as$.

So is it right to the proof this way?
 
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  • #2
Cbarker1 said:
Dear Everyone,

Here is the question:

"Prove that if $k$ divides the integers $a$ and $b$, then $k$ divides $as+bt$ for every pair of integers $s$ and $t$ for every pair of integers."

The attempted work:

Suppose $k$ divides $a$ and $k$ divides $b$, where $a,b\in\mathbb{Z}$. Then, $a=kt$ and $b=ks$, where $s,t\in\mathbb{Z}$ (Here is where I am stuck).

Do I solve for $k$?

If I do solve for $k$, then it yields the
$k=\frac{a}{t}$. Then, $b=\frac{as}{t}$. So $bt-as=0$. Then $0$ divides $bt-as$.

So is it right to the proof this way?
Hi Cbarker1,

$s$ and $t$ are arbitrary integers in the question, you should not use them as you do in the proof, where they are fixed integers that depend on $a$, $b$, and $k$.

You can say that there are integers $m$ and $n$ such that $a=km$ and $b=kn$. Now, you have:
$$
as+bt = (km)s + (kn)t = k(ms+nt)
$$
As $ms+nt$ is an integer, this shows that $k$ divides $as+bt$.
 

FAQ: Elementary Number Theory proof

What is Elementary Number Theory?

Elementary Number Theory is a branch of mathematics that deals with the properties and relationships of integers. It focuses on understanding the fundamental concepts of numbers, such as prime numbers, divisibility, and congruence.

What is a proof in Elementary Number Theory?

A proof in Elementary Number Theory is a logical argument that uses mathematical principles and reasoning to demonstrate the truth of a statement or theorem. It involves breaking down a problem into smaller, more manageable steps and providing evidence for each step to show that the statement is true for all cases.

Why are proofs important in Elementary Number Theory?

Proofs are important in Elementary Number Theory because they allow us to establish the truth of mathematical statements and theorems. They also help us to understand the underlying principles and patterns in numbers, and can be used to solve more complex problems in mathematics and other fields.

What are some common techniques used in Elementary Number Theory proofs?

Some common techniques used in Elementary Number Theory proofs include mathematical induction, direct proof, proof by contradiction, and proof by contrapositive. These techniques involve using logical reasoning, algebraic manipulation, and properties of numbers to demonstrate the truth of a statement.

How can I improve my skills in writing Elementary Number Theory proofs?

To improve your skills in writing Elementary Number Theory proofs, it is important to practice regularly and familiarize yourself with different proof techniques. You can also read and study the proofs of established theorems to gain a better understanding of how to structure and present your own proofs. Seeking guidance and feedback from a mentor or teacher can also be helpful in improving your proof writing skills.

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