If g c d ( a , c ) = 1 and g c d ( b , c ) = 1, prove that g c d ( a b , c ) = 1

Question

If   g  c  d  (  a  ,  c  )  =  1 and   g  c  d  (  b  ,  c  )  =  1, prove that   g  c  d  (  a  b  ,  c  )  =  1

lelapem · Accepted Answer

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:

(a k + c l) \cdot (b m + c n) = 1 \cdot 1 a b k m + a c k n + c b l m + c c l n = 1

or

a b (k m) + c (a k n + b l m + c l n) = 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