
[SOLVED] Coprime
Hi, i'm having trouble with the following questions;
Let a, b be nonzero integers.
a) Show that if there are x,y in Z (integers) with ax + by =1, then a and b are coprime.
b) Show that the integers a/(a,b) and b/(a,b) are coprime.
Thanks for you help.

1) Let d such as $\displaystyle da$ and $\displaystyle db$
Then $\displaystyle dax+by\Rightarrow d1\Rightarrow d=1$
So, a and b are coprime.
2) Let $\displaystyle d=(a,b)$
Then $\displaystyle a=da_1, \ b=db_1$ with $\displaystyle (a_1,b_1)=1$
But $\displaystyle \frac{a}{d}=a_1, \ \frac{b}{d}=b_1$, so $\displaystyle \frac{a}{d}, \ \frac{b}{d}$ are coprime.