Given three integers a, b and c, we have:
(a ∧ b = 1 and a | bc) ⇒ a | c.
Demonstration :
Suppose a ∧ b = 1 and a | bc. According to the Bezout identity, there are two integers u and v such that au + bv = 1, which implies acu+bcv= c.
as a | bc, a | bcv and therefore:
a | acu + bcv=c.
(a ∧ b = 1 and a | bc) ⇒ a | c.
Demonstration :
Suppose a ∧ b = 1 and a | bc. According to the Bezout identity, there are two integers u and v such that au + bv = 1, which implies acu+bcv= c.
as a | bc, a | bcv and therefore:
a | acu + bcv=c.
What a great theorem !
ReplyDelete