Show that there exists integers $w,z$ such that $\gcd(a,b)w+cz=e$

Question

Let $\gcd(a,b,c)\mid e$ . Show that there exists integers $w,z$ such that $\gcd(a,b)w+cz=e$

Assume that $\gcd(a,b)=d$ then $a=dp,b=dq$.

Also there exists integers $m,n$ such that $am+nb=d$.

How to find integers $w,z$ such that $dw+cz=e$?

Can someone please give some hint?

Hint $\ $ GCD is Associative so $\ ((a,b),c) = (a,b,c)\ \ $ — Bill Dubuque, Aug 16 '19 at 14:50

score 2 · Accepted Answer · answered Aug 16 '19 at 14:30

2

It is known that (see wiki-page) the linear Diophantine equation $dw+cz=e$ has a solution $(w,z)$ if and only if $\gcd(d,c)$ divides $e$. But this is true because $$\gcd(d,c)=\gcd(\gcd(a,b),c)=\gcd(a,b,c).$$

answered Aug 16 '19 at 14:30

Robert Z

145,942

I dont get it.can you elaborate? – Charlotte Aug 16 '19 at 14:33
What is your doubt? – Robert Z Aug 16 '19 at 14:34
Why is $\gcd(\gcd(a,b),c)=\gcd(a,b,c)$ – Charlotte Aug 16 '19 at 14:50
This is a known property of $\gcd$. See the 11th one here: https://en.wikipedia.org/wiki/Greatest_common_divisor#Properties – Robert Z Aug 16 '19 at 14:55
Oh okay,thank you so much sir – Charlotte Aug 16 '19 at 14:57
You are welcome! – Robert Z Aug 16 '19 at 14:57
Can you please help me with https://math.stackexchange.com/q/3325185/665065 – Charlotte Aug 16 '19 at 15:07

Show that there exists integers $w,z$ such that $\gcd(a,b)w+cz=e$

1 Answers1