Question

Provide the complete solution to:

3x - 2 ≡ 7 (mod 11)
5 ≡ 4x - 1 (mod 9??)

Answer 1

\[3x-2 \equiv 7 (\mod 11) => 3x-2-7=11k \text{ for some integer k}\]
\[3x-9=11k\]
\[3x=11k+9\]
k=0 => x=3
k=1 => x=not integer
k=2 => x=not integer
k=3 => x=14
k=4 => x=not integer
k=5 => x=not integer
k=6 => x=25
k=7 => x=not integer
k=8 => x=not integer
x=9 => x=36
so we have x=3,14,25,36,...
x=11i+3 for i=0,1,2,3,4,....

Answer 2

above just say where i is integer
you know so we can include all integers not just the positives
\[5 \equiv 4x-1 (\mod 9) =>5-(4x-1) = 9k \text{ for some integer k}\]
\[-4x+6 =9k\]
\[4x=6-9k\]

k=-2=>x=6
k=-1=>x=not integer
k=0=> x=not integer
k=1=> x=not integer
k=2=> x=-3
k=3=> x=not integer
k=4=> x=not integer
k=5=> x=not integer
k=6=> x=-12
k=7=> x=not integer
k=8=> x=not integer
k=9=> x=not integer
k=10=>x=-21
k=11=>not integer

x=6,-3,-12,-21,...

x=42-9(4+i) where i is integer