Answer to Question #139165 in Abstract Algebra for J

Question

Assume that a | b times c and that gcf(a,b)=1. Prove that a | c(hint: use the result that gcf(a,b)=1 iff there exist x,y is an element of Z such that a times x+b times y=1)

Answer 1

We have given that $a|bc$ and gcf(a,b)=1.

Thus there exist integers x,y such that

$$ax+by=1\hspace{1cm}(1)$$

and, as $a|bc$ , there exist integer $k$ such that

$$ak=bc\hspace{1cm}(2)$$

Now, multiply $c$ in both side of the equation (1), we get

$$acx+bcy=c$$

Now, using equation (2) we get

$$acx+aky=c\\ \implies a(cx+ky)=c\\ \implies a|c$$