If we know $\gcd(a,b,c)=1$, does it mean $\gcd\left(\frac{a}{\gcd\left(a,b\right)},\gcd\left(c,\frac{b}{\gcd\left(a,b\right)}\right)\right)=1$

Question

If we know $gcd(a,b,c)=1$, how to prove that $gcd\left(\frac{a}{gcd\left(a,b\right)},gcd\left(c,\frac{b}{gcd\left(a,b\right)}\right)\right)=1$? Where $a, b, c$ are any positive integers.

I have test two specific examples, looks it is true:

Example 1): Such as when $a=35,b=55,c=77\Longrightarrow\ gcd(35,55,77)=1$, we have $gcd\left(\frac{35}{gcd\left(35,55\right)},gcd\left(77,\frac{55}{gcd\left(35,55\right)}\right)\right)=gcd\left(7,gcd\left(77,11\right)\right)=gcd(7,11)=1$

Example 2): Such as when $a=3,b=5,c=7\Longrightarrow\ gcd(3,5,7)=1$, we have $gcd\left(\frac{3}{gcd\left(3,5\right)},gcd\left(7,\frac{5}{gcd\left(3,5\right)}\right)\right)=gcd\left(3,gcd\left(7,1\right)\right)=gcd(3,1)=1$

By the linked dupes $,d=(a,b) \Rightarrow (a/d,(c,b/d)) = ((a/d,b/d),c) = (1,c)=1\ \ $ — Bill Dubuque, Aug 28 '23 at 17:15

score -1 · Accepted Answer · answered Aug 28 '23 at 14:53

-1

We are gonna use these two facts: $gcd(x,gcd(y,z)) = gcd(gcd(x,y),z) = gcd(x,y,z)$ and for every positive integer w, $gcd(w*x,w*y) = w*gcd(x,y)$

Let $g = gcd(a,b)$

$g * gcd(a/g,b/g) = gcd(g * a/g,g * b/g) = gcd (a,b) = g$

Cancelling g from both sides, $gcd(a/g,b/g)=1$

Let $k = gcd(a/g , gcd(c , b/g))$

$k = gcd(a/g , gcd(b/g , c)) = gcd( gcd(a/g,b/g), c)) = gcd( 1, c) = 1$

answered Aug 28 '23 at 14:53

Hussein Aiman

392

Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. – José Carlos Santos Aug 28 '23 at 14:57
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Please strive not to post more (dupe) answers to dupes of FAQs, cf. recent site policy announcement here. – Bill Dubuque Aug 28 '23 at 17:15
Further, please do not duplicate prior-posted answers on this page. – Bill Dubuque Aug 28 '23 at 17:57
I didn't duplicate any prior-posted answers on this page. I only saw the other answer after I finished writing mine, and the two answers should have been written a few minutes apart if you would check that @BillDubuque – Hussein Aiman Aug 28 '23 at 18:07
The other answer was posted 20 minutes prior, not "a few minutes apart". – Bill Dubuque Aug 28 '23 at 19:32
That's a few minutes apart by my standards. Point is that the question was unanswered on my screen at the time I posted my answer, until the page reloaded. Not sure if there is any benefit from arguing about that – Hussein Aiman Aug 28 '23 at 22:30
If you take a long time after loading the page before posting an answer then you should always refresh the page to avoid posting duplicate answers. – Bill Dubuque Aug 28 '23 at 23:32
Okay, I'll keep that in mind next time – Hussein Aiman Aug 28 '23 at 23:42

If we know $\gcd(a,b,c)=1$, does it mean $\gcd\left(\frac{a}{\gcd\left(a,b\right)},\gcd\left(c,\frac{b}{\gcd\left(a,b\right)}\right)\right)=1$

1 Answers1