Recall that a congruence is an equation of the form
; a linear congruence should be that equation with
P(x)=ax-b - and it is!
which means that ax-b=ny for some 
; rewriting, we have that
to be solved in integers - that is, a Diophantine equation!
Now the Diophantine equation could have an infinite number of solutions, but since we're working modulo n, we're only interested in solutions distinct mod n.
linear congruence a congruence in which P(x) is of the form P(x)=ax-b.
Theorem 4.7: The linear congruence 
has a solution if
and only if d | b, where 
. If d|b, then the linear
congruence has d mutually incongruent solutions modulo n.
Corollary:
If gcd(a,n)=1, then the linear congruence
has a unique solution modulo n.
Example #1bdf, p. 82: Solve the following linear congruences:
-
-
-
[Hint:
]
Theorem 4.8 (The Chinese Remainder Theorem): Let 
be
positive integers such that 
for 
. Then the
system of linear congruences
has a simultaneous solution which is unique modulo 
.
The unique solution is of the form
where 
and 
is the unique solution to the linear
congruence
.
Example: Find x such that x hours from midnight will be 6:00 AM, and such that x days from Sunday will be Thursday.
Theorem 4.9: The system of linear congruences
has a unique solution whenever 
.
For those of you with linear algebra backgrounds: ad-bc in the linear system of Theorem 4.9 you'll recognize as the determinant.
Example #20, p. 83: Find the solutions of each:
- (a)
- (b)
- (c)
