How do i prove that $\gcd(a_1,\ldots,a_n)\operatorname{lcm}(a_1,\ldots,a_n)=a_1\cdots a_n$?

Asked Apr 13 '14 at 14:06

Active Apr 13 '14 at 14:38

Viewed 184 times

Let $a_1,\ldots,a_n$ be nonzero integers. Define $$G=\{A\in\mathbb{Z}:A \text{ is a linear combination of } a_1,\ldots a_n\}$$

My definition for $\gcd(a_1,\ldots,a_n)$ is the principal ideal of $G$.

Define $G_i$ be the ideal generated by $a_i$.

My definition for $\operatorname{lcm}(a_1,\ldots,a_n)$ is the principal ideal of $\bigcap_{i=1}^n G_i$

(This is Andre Weil's definition)

With this definition, how do i prove $$\gcd(a_1,\ldots,a_n)\operatorname{lcm}(a_1,\ldots,a_n)=a_1\cdots a_n$$

edited Apr 13 '14 at 14:38

Michael Hardy

asked Apr 13 '14 at 14:06

user140374

1

try with $$15,35,105$$ – lab bhattacharjee Apr 13 '14 at 14:07
@labbhattacharjee Then, what restriction on $n$ would make this hold? At least, how do i prove this when $n=2$? – user140374 Apr 13 '14 at 14:09
http://www.proofwiki.org/wiki/Product_of_GCD_and_LCM – lab bhattacharjee Apr 13 '14 at 14:10
@labbhattacharjee Thank you! Still, are multiple GCD,LCM not really necessary in general? I remember that multiple GCD,LCM are necessary in group theory for counting order of a cyclic groups.. – user140374 Apr 13 '14 at 14:15
http://math.stackexchange.com/questions/319297/gcd-to-lcm-of-multiple-numbers – lab bhattacharjee Apr 13 '14 at 14:18
http://www.cut-the-knot.org/arithmetic/GcdLcmProperties.shtml – lab bhattacharjee Apr 13 '14 at 14:20
@labbhattacharjee I don't get the last assertion in proofwiki. I have showed that |ab|≤gcd(a,b)lcm(a,b) , but i don't get why the equality holds – user140374 Apr 13 '14 at 15:17
http://www.cut-the-knot.org/Curriculum/Arithmetic/GcdLcm.shtml – lab bhattacharjee Apr 13 '14 at 15:18

0 Answers0