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### 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 ...
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### $n$ 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
### 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 ...