Proving ab|c: The Proof for (a,b)=1

[chaotixmonjuish]

If a|c and b|c with (a,b)=1, prove ab|c

The book just states that ab|c if (a,b)=1...so I took a stab on proving it:

(a,b)=1 means au+bv=1

so for no reason at all I threw in a c

acu+bcv=c

since a|c and b|c c=ak and c= bh

abhu+bakv=c

this means ab(hu+kv)=c

hence ab|c
It this proof right...the book kind of skips proving this proposition.

 

[Take_it_Easy]

Yes that's RIGHT!

 

[chaotixmonjuish]

Great! The book simply told me that the proposition is possible only because of the GCD...and the lack of a proof bothered me.

 

[Take_it_Easy]

I agree with you, books should be more detailed!
By the way "congratulations!" since you proved very good in finding the proof by yourself!
Are you studing Algebra alone by yourself? Or are you attending university?

 

[chaotixmonjuish]

I'm actually taking an Intro to Abstract type course and I'm just aggressively nuturing my curiosity by borrowing abstract algebra books from the library and working stuff out.