-4 0 mod 4
3 1 mod 4
2 2 mod 4
1 3 mod 4
0 0 mod 4
1 1 mod 4
2 2 mod 4
3 3 mod 4
4 0 mod 4
-What are some examples of congruences in real life?
-On a clock, the time is shown modulo 12. Thus the hour can be 0, 1, 2, . . . , 11,
-An integer p > 1 is called a prime if its only positive divisors are 1 and p;
otherwise it is called composite.
-Prime numbers are important because they are the building blocks of the integers. Every
integer greater than 1 can be written a
Equivalence Class Properties
-Let R be any equivalence relation on the set S. If a, b S, then
[a] = [b] iff (a, b) R
[a] [b] = iff (a, b) / R
-The set of congruence classes of integers under the congruence relation modulo m is called the set of inte
-Let A be a set and R be a relation on A. If R is reflexive, symmetric, and
transitive, then we say that R is an equivalence relation.
-Let R be an equivalence relation on the set A, and let x A. Then the
equivalence class of x is the se
-Congruences are also useful in expressing some of our earlier results. Recall Bezouts
Identity (Theorem 10) which considered whether integers x and y exist which satisfy a
Diophantine equation of the form
ax + by = c
for some gi
GCD and Prime Numbers
-If a = 2^6 3^4 and b = 2^3 3^2 5^2, then gcd(a, b) = 2^3 3^2.
-prove that if a|n and b|n, then ab|n using the Unique Factorization Theorem
Let P = p1, p2, . . . , pn denote the unique set of primes in the prime factorization of a. L
Unique Prime Factorization Theorem
-We have already seen in Theorem 12 that every integer greater than 1 can be expressed
as a product of primes. We are heading towards the Fundamental Theorem of Arithmetic
(FTOA), also called the Unique Prime Factorizati
-Let A and B be sets. A relation between A and B is a set of ordered pairs,
R A B. Given a A and b B, we say that a and b are related iff (a, b) R. It is
also common to write xRy to mean (x, y) R.
If A = B, we say that R is a relation on A.
Properties of Modulus
-Let a, b Z.
a + b (a mod m) + (b mod m) mod m
a b (a mod m) (b mod m) mod m
ab (a mod m) (b mod m) mod m
The notation a mod m above is used to denote that unique remainder, r, where
r a mod m and 0 r < m.
-What is the remainder when
-If a, b Z, where a > 0, then unique q, r Z, s.t.,
b = qa + r, where 0 r < a
Here q is called the quotient of b divided by a, and r is called the remainder of b divided
-Given a fixed positive integer m > 0. If a, b Z, we say that a is c