If g c d ( a , c ) = 1 and g c d...

Marcelo Mullins

Marcelo Mullins

Answered

2022-07-26

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

Answer & Explanation

lelapem

lelapem

Expert

2022-07-27Added 12 answers

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 ) ( b m + c n ) = 1 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

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