# Diophantine equations

• Oct 12th 2009, 01:48 AM
manusform
Diophantine equations
Hi this is a hard thing. Solving Diophantine equations with many variables. My mind hang with this. Somebody helps me. This is it:

x+y congruence to 1(mod7)
x+z congruence to 2(mod7)
y+z congruence to 3(mod7)
• Oct 12th 2009, 03:48 AM
tonio
Quote:

Originally Posted by manusform
Hi this is a hard thing. Solving Diophantine equations with many variables. My mind hang with this. Somebody helps me. This is it:

x+y congruence to 1(mod7)
x+z congruence to 2(mod7)
y+z congruence to 3(mod7)

Write x + y = 1 + 7k , x + z = 2 + 7m , y + z = 3 + 7n , k,m,n integers.
Substract now the 3rd. one from the 1st.:

x - z = -2 + 7(k-n) , and now add this to the second one

2x = 7(k-n+m) ==> x = 0 (mod 7), and from here y = 1 (mod 7), z = 2(mod 7), and the general solution is:

x = 7k
y = 1 + 7m
z = 2 + 7n

k,m,n integers.

Tonio
• Oct 12th 2009, 06:04 PM
Media_Man
x+y=1, x+z=2, y+z=3, so x+y+z=(1+2+3)/2=3. Since y+z=3, x=0. It follows that y=1 and z=2. (The same solution holds in modulo for any base.)