algebra of remainders

• Oct 17th 2006, 02:37 AM
edgar davids
algebra of remainders
can some1 pls help noone can do this proof and preventing me fully understanding the chinese remainder thm

Edgar

z = integers

prove the following generalisation of the Chinese Remainder Theorem. Show that if m1,m2,m3, are pairwise coprime integers and a1 is an element of z/m1 and a2 is an element of z/m2 and a3 is an element of z/m3 then there is a unique x element of z/(m1m2m3) such that

x identical a1 mod m1
x identical a2 mod m2
x identical a3 mod m3
• Oct 20th 2006, 09:29 AM
ThePerfectHacker
Quote:

Originally Posted by edgar davids
can some1 pls help noone can do this proof and preventing me fully understanding the chinese remainder thm

Edgar

z = integers

prove the following generalisation of the Chinese Remainder Theorem. Show that if m1,m2,m3, are pairwise coprime integers and a1 is an element of z/m1 and a2 is an element of z/m2 and a3 is an element of z/m3 then there is a unique x element of z/(m1m2m3) such that

x identical a1 mod m1
x identical a2 mod m2
x identical a3 mod m3

The basic theorem (Chinese Remainder) says if you are given a system:
$x\equiv a_1 (\mbox{mod } m_1)$
$x\equiv a_2 (\mbox{mod } m_2)$
................................
$x\equiv a_n (\mbox{mod } m_n)$
Then there is a unique x that satisfyies all of these congruences.