Answer to Question #139165 in Abstract Algebra for J

Question #139165
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)
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Expert's answer
2020-10-19T17:53:41-0400

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

Thus there exist integers x,y such that

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

and, as abca|bc , there exist integer kk such that

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

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


acx+bcy=cacx+bcy=c

Now, using equation (2) we get


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


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