Linked Questions

5
votes
2answers
376 views

A Proof of the Fundamental Theorem of Arithmetic [duplicate]

Is there a proof of the Fundamental Theorem of Arithemetic that does not make use of the Integers or Rational Numbers (as opposed to using only the Natural Numbers)? And if so, what is it? By the ...
0
votes
2answers
48 views

$n$ is prime $\Leftrightarrow [a][b]=0\Rightarrow [a]=[b]=0 $ [duplicate]

I'm stuck on the following exercise from Herstein's "Topics in Algebra": "show that ($n$ is prime) $\Leftrightarrow ([a][b]=[0]\Rightarrow [a]=[b]=[0]) $ in $J_n$". for the rightward implication I ...
47
votes
6answers
20k views

Why are maximal ideals prime?

Could anyone explain to me why maximal ideals are prime? I'm approaching it like this, let $R$ be a commutative ring with $1$ and let $A$ be a maximal ideal. Let $a,b\in R:ab\in A$. I'm trying to ...
20
votes
5answers
6k views

Solving linear congruences by hand: modular fractions and inverses

When I am faced with a simple linear congruence such as $$9x \equiv 7 \pmod{13}$$ and I am working without any calculating aid handy, I tend to do something like the following: "Notice" that adding $...
24
votes
3answers
23k views

If $p$ is a prime and $p \mid ab$, then $p \mid a$ or $p \mid b$.

The proof is already given in the textbook but I tried other way around. Proof by contradiction: Let's assume that $p$ doesn't divide $a$ and $p$ doesn't divide $b$, but $p$ divides $ab$. So $\gcd(p,...
6
votes
10answers
1k views

If a prime $p\mid ab$, then $p\mid a$ or $p\mid b\ $ [Euclid's Lemma]

If a prime number $p$ is a divisor of a product $ab$, $p$ has to be a divisor of $b$ or $a$. How can I demonstrate this theorem? I demonstrated this theorem on one way using Bezout's theorem in an ...
5
votes
4answers
3k views

Not understanding Simple Modulus Congruency

Hi this is my first time posting on here... so please bear with me :P I was just wondering how I can solve something like this: $$25x ≡ 3 \pmod{109}.$$ If someone can give a break down on how to do ...
4
votes
4answers
7k views

Prove that $(ma, mb) = |m|(a, b)\ $ [GCD & LCM Distributive Law]

I'm trying to prove that $(ma, mb) = $|$m$|$(a, b)$ , where $(ma, mb)$ is the greatest common divisor between $ma$ and $mb$. My thoughts: If $(ma, mb) = d$ , then $d$|$ma$ and $d$|$mb$ → $d$|$max + ...
3
votes
2answers
104 views

Prove that if $p\mid ab$ where $a$ and $b$ are positive integers and $a\lt p$ then $p\le b$

I have found an old textbook called "Real Variables by Claude W. Burrill and John R. Knudsen" in the first chapter this textbook uses 15 axioms to derive much of the well known and basic facts about ...
1
vote
3answers
195 views

If $19|x^2$ then $19|x$ [duplicate]

$x\notin 19\mathbb{Z}$ If $19|x^2$ then $19|x$. Proof by contraposition: If $19\not|x$, then $19\not|x^2$ If $19\not|x$ then $x$ must take on one of the following forms: $x=19k+1, x=19+2, x=19k+3,...
1
vote
2answers
64 views

Prove prime $p\mid ab\,\Rightarrow\, p\mid a\,$ or $\,p\mid b\,$ without using Fundamental Theorem of Arithmetic

Let: $p$ $\in \mathbb{P}$ $\wedge$ $n_{1},n_{2}\in \mathbb{Z}$. Then: $p|(n_{1}n_{2})\implies p|n_{1} \vee \space p|n_{2} $ This little hypothesis is straightforward while using fundamental theorem ...