Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
1  
The second is a typo for max –  André Nicolas Jan 30 '13 at 20:32
1  
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
1  
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
add comment

1 Answer

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 $\mathbb{C}$.

share|improve this answer
    
Thought so. Thank you. –  Kaish Jan 30 '13 at 20:34
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.