Marcelo Mullins

2022-07-26

If $gcd\left(a,c\right)=1$ and $gcd\left(b,c\right)=1$, prove that $gcd\left(ab,c\right)=1$

lelapem

Expert

to prove that gcd(ab, c) = 1, we need to show that there exists integers x and y such that abx+cy=1 ( using property of relatively prime)
given that gcd(a, c) = 1
so there exists some integers k and l such that
ak+cl=1 ( using property of relatively prime)
given that gcd(b, c) = 1
so there exists some integers m and n such that
bm+cn=1 ( using property of relatively prime)
multiply both equations we get:
$\left(ak+cl\right)\cdot \left(bm+cn\right)=1\cdot 1\phantom{\rule{0ex}{0ex}}abkm+ackn+cblm+ccln=1$
or
$ab\left(km\right)+c\left(akn+blm+cln\right)=1$
or abx+cy=1 where x=km and y=akn+blm+cln
because product and sum of integers also give integer.
as we have proved that abx+cy=1, Hence gcd(ab, c) = 1

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