PMATH340 LN8 - Chapter 4 Congruences For this section it is...

Chapter 4
Congruences
For this section, it is helpful if you keep “clock arithmetic” in mind. That is
14:00 hours is 2 o’clock. This will be considered doing arithmetic modulo 12.
We will just consider an n hour clock.
Definition 5. Let n ∈ N . Two integers a and b are said to be congruent modulo
n , written
a ≡ b (mod n ) ,
provided n | ( a − b ).
The canonical example of a congruence comes from the division algorithm.
Example 5. Let a ∈ Z and n ∈ N . Given the division algorithm we know that
there exists unique q, r ∈ Z with 0 ≤ r < n , such that
a = qn + r.
The definition of congruence gives
a ≡ r (mod n ) .
Definition 6. If a, b ∈ Z and a ≡ b (mod n ) we say that a is a residue of b
modulo n .
Remark 4. The above examples tells us that modulo n any integer a is con-
gruent to one of
0 , 1 , 2 , . . . , n − 1 .
These integers are called the least nonnegative residues modulo n .
Definition 7. The set of integers { a
0
, a
1
, . . . , a
n − 1
} is called a complete residue
system modulo n provided
(i) a
i
≡ a
j
(mod n ), whenever i = j
(ii) for each integer k there corresponds an a
i
such that k ≡ a
i
(mod n ).