If $\gcd (a, bc) = d$ and $\gcd(a, b) = 1$, then $\gcd(a, c) = d $

Question

I have proceeded to as far as declaring these with EEA but I do not know how to show that $d \mid c$ for the conclusion. Any help to go to this direction will be greatly appreciated.

Three answers, two down-votes, and I'm the only one who's up-voted the question so far, even though I haven't answered it. Why was it downvoted? $\qquad$ — Michael Hardy, Jun 22 '16 at 22:08
No idea. I was going to comment to ask what "declaring these with EEA" actually meant in practical terms. - so @Reginsmal can you enlighten me? — Joffan, Jun 22 '16 at 22:36

score 1 · Answer 1 · answered Jun 22 '16 at 21:40

1

Use Gauß's lemma:

If $d\mid bc$ and $\gcd(d,b)=1$, then $d\mid c$.

answered Jun 22 '16 at 21:40

Bernard

175,478

Please see my comment under the question. $\qquad$ – Michael Hardy Jun 22 '16 at 22:08
@Michael Hardy: I feel like here are some temperamental people on this site… I've added my own upvote. – Bernard Jun 22 '16 at 22:17

score 1 · Accepted Answer · edited Jun 22 '16 at 22:06

1

Note that $d \mid a$ because $\gcd(a,bc) = d$.

We need $d \mid c$ as well, as you state. We know that $d \mid bc$ so that $bc = Ad$ for some $A$.

Then there are $C,D$ such that $Ca + Db = 1$, by $\gcd(a,b) = 1$.

So $Cac + Dbc = c$ and so $Cac + DAd = c$, using $bc = Ad$.

As $d \mid a$, $d$ divides $Cac$ and $DAd$, and hence their sum as well. So $d$ divides $c$, as required.

edited Jun 22 '16 at 22:06

Michael Hardy

1

answered Jun 22 '16 at 21:45

Henno Brandsma

242,131

Please see my comment under the question. $\qquad$ – Michael Hardy Jun 22 '16 at 22:08

score 1 · Answer 3 · answered Jun 22 '16 at 22:05

1

We know that $d \mid a$. However, since $\gcd(a,b) = 1$, we also know that $b$ and $d$ share no common factors (that is, $\gcd(d,b) = 1$).

Then, since we also know that $d \mid bc$, it must also be that $d \mid c$.

answered Jun 22 '16 at 22:05

Joffan

39,627

Please see my comment under the question. $\qquad$ – Michael Hardy Jun 22 '16 at 22:08
@MichaelHardy Not guilty of a downvote :-) - is there some convention that says anyone answering should upvote the question? – Joffan Jun 22 '16 at 22:37
I don't think it's a convention, but it seems to me a question worth answering is almost always worth up-voting. $\qquad$ – Michael Hardy Jun 22 '16 at 22:40

If $\gcd (a, bc) = d$ and $\gcd(a, b) = 1$, then $\gcd(a, c) = d $

3 Answers3