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I'm just going through my lecture notes and under one subheading "application", I have written:

If $m = \prod_{i = 1}^k p_i^{r_i}$ and $n = \prod_{i = 1}^k p_i^{s_i}$, for primes $p_i$ & $r_i, s_i \in \mathbb{N}$. Then $\gcd(m,n) = \prod_{i = 1}^k p_i^{\min(r_i,s_i)}$ and $\mathrm{lcm}(m,n) = \prod_{i = 1}^k p_i^{\min(r_i,s_i)}$.

Should they both be $\min(r_i,s_i)$, because this just gives gcd = lcm doesn't it?

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The second is a typo for max – André Nicolas Jan 30 '13 at 20:32
Do you mean the fundamental theorem of arithmetic? – Jonas Meyer Jan 30 '13 at 20:34
@JonasMeyer I swear my lecturer said that but then she (definitely) wrote "algebra" on the board, so I just thought maybe it has two names? – Kaish Jan 30 '13 at 20:37
Just googled Fundamental Theorem of Algebra and it is nothing like theory of arithmetic lol. – Kaish Jan 30 '13 at 20:38
@Kaish: Please: TeX is sophisticated. Write \prod_{i=1}^k, not \Pi_{i=1}^k. These don't look the same $\displaystyle\prod_{i=1}^k$ versus $\displaystyle\Pi_{i=1}^k$. Also, don't write \mathrm{min}; just write \min. Look: $\displaystyle\min_{a\in A} f(a)$ versus $\displaystyle\mathrm{min}_{a\in A} f(a)$. The latter form also fails to provide proper spacing before and after, as in $a\min b$. And \gcd is also a standard operator name in LaTeX. – Michael Hardy Jan 30 '13 at 21:24
up vote 2 down vote accepted

The second one is a typo, it should read $\max$.

Remark: The factorization result mentioned in the post is called the Fundamental Theorem of Arithmetic, or the Unique Factorization Theorem.

The Fundamental Theorem of Algebra has various statements. One of them is that a non-constant polynomial with coefficients in $\mathbb{C}$, the set of complex numbers, always has a root in $\mathbb{C}$.

It can be viewed as a factorization theorem, because it is equivalent to the result that a polynomial of degree $\ge 1$ over $\mathbb{C}$ can be expressed as a product of linear polynomials with coefficients in $\mathbb{C}$.

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Thought so. Thank you. – Kaish Jan 30 '13 at 20:34

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