Things are getting easy peasy as long as thinking is learnt. Such easy peasiness in things are initially rooted in easy peasy math.
Sunday, October 7, 2012
Integers modulo m)
It has been kindly ackhnowledged that this lecture was downloaded from the website:http://home.scarlet.be/math for some students learning the congruences.
Modular arithmetic is quite a useful tool in number theory. In particular, it
can be used to obtain information about the solutions (or lack thereof) of a
specific equation. This page gives a fairly detailed introduction. Another good
introduction, in the form of an interactive tutorial, can be found in Part 2 of
Math
Alive: Cryptography.
Contents
Everyone knows that set of integers can be broken up into the following two
classes:
the even numbers (..., –6, –4, –2, 0, 2, 4, 6,...); and
the odd numbers (..., –5, –3, –1, 1, 3, 5,...).
There are certain generalizations we can make about the arithmetic of numbers
based on which of these two classes they come from. For example, we know that
the sum of two even numbers is even. The sum of an even number and an odd number
is odd. The sum of two odd numbers is even. The product of two even numbers is
even, etc.
Modular arithmetic lets us state these results quite precisely, and it also
provides a convenient language for similar but slightly more complex statements.
In the above example, our modulus is the number 2. The modulus can be
thought of as the number of classes that we have broken the integers up into. It
is also the difference between any two "consecutive" numbers in a given class.
Now we represent each of our two classes by a single symbol. We let the
symbol "0" mean "the class of all even numbers" and the symbol "1" mean "the
class of all odd numbers". There is no great reason why we have chosen the
symbols 0 and 1; we could have chosen 2 and 1, or –32 and 177, but 0 and 1 are
the conventional choices.
The statement "the sum of two even numbers is even" can be expressed by the
following:
0 + 0 ≡ 0 mod 2.
Here, the "≡" symbol is not equality but congruence, and the "mod 2"
just signifies that our modulus is 2. The above statement is read "Zero plus
zero is congruent to zero, modulo two." The statement "the sum of an even number
and an odd number is odd" is represented by
0 + 1 ≡ 1 mod 2.
Those examples are natural enough. But how do we write "the sum of two odd
numbers is even"? It is the (at first strange looking) expression
1 + 1 ≡ 0 mod 2.
Here the symbols "≡" and "mod 2" are suddenly very important! We have
analogous statements for multiplication:
0 × 0 ≡ 0 mod 2, 0 × 1 ≡ 0 mod 2, 1 × 1 ≡ 1 mod 2.
In a sense, we have created a number system with addition and multiplication
but in which the only numbers that exist are 0 and 1. You may ask what use this
has. Well, our number system is the system of integers modulo 2, and
because of the previous six properties, any arithmetic done in the integers
translates to arithmetic done in the integers modulo 2. This means that if
we take any equality involving addition and multiplication of integers, say
12 × 43 + 65 × 78 = 5586,
then reducing each integer modulo 2 (i.e. replacing each integer by
its class "representative" 0 or 1), then we will obtain a valid congruence. The
above example reduces to
0 × 1 + 1 × 0 ≡ 0 mod 2,
or 0 + 0 ≡ 0 mod 2.
More useful applications of reduction modulo 2 are found in solving
equations. Suppose we want to know which integers might solve the equation
3a – 3 = 12.
Of course, we could solve for a, but if we didn't need to know what
a is exactly and only cared about, say, whether it was even or odd, we
could do the following. Reducing modulo 2 gives the congruence
1a + 1 ≡ 0 mod 2,
or
a ≡ –1 ≡ 1 mod 2,
so any integer a satisfying the equation 3a – 3 = 12 must be
odd.
Since any integer solution of an equation reduces to a solution modulo 2, it
follows that if there is no solution modulo 2, then there is no solution in
integers. For example, assume that a is an integer solution to
2a – 3 = 12,
which reduces to
0 · a + 1 ≡ 0 mod 2,
or 1 ≡ 0 mod 2. This is a contradiction because 0 and 1 are different numbers
modulo 2 (no even number is an odd number, and vice versa). Therefore the above
congruence has no solution, so a couldn't have been an integer. This
proves that the equation 2a – 3 = 12 has no integer solution.
Less trivially, consider the system of equations
6a – 5b = 4, 2a + 3b = 3.
Modulo 2, these equations reduce to
0 + 1b ≡ 0 mod 2, 0 + 1b ≡ 1 mod 2.
This says that b is both even and odd, which is a contradiction.
Therefore we know that the original system of equations has no integer
solutions, and to prove this we didn't even need to know anything about
a.
As shown by the preceding examples, one of the powers of modular arithmetic
is the ability to show, often very simply, that certain equations and systems of
equations have no integer solutions. Without modular arithmetic, we would have
to find all of the solutions and then see if any turned out to be integers.
II. Definition and Further Examples
Of course, there is nothing special about the number 2. Any integer (except
0) will work for the modulus m. We now give the mathematical definition
of congruence.
Definition. Let m ≠ 0 be an integer. We say that two integers
a and b are congruent modulo m if there is an integer
k such that a – b = km, and in this case we write
a ≡ b mod m.
Notice that the condition "a – b = km for some integer
k" is equivalent to the condition "m divides a – b".
In the previous section we used the modulus m = 2. Although we wrote
congruences using only 0 and 1, really any integers are valid. Modulo 2, all of
the even numbers are congruent to each other since the difference of any two
even numbers is divisible by 2:
... ≡ –6 ≡ –4 ≡ –2 ≡ 0 ≡ 2 ≡ 4 ≡ 6 ≡ ... mod 2.
Also, every odd number is congruent to every other odd number modulo 2 since
the difference of any two odd numbers is even:
... ≡ –5 ≡ –3 ≡ –1 ≡ 1 ≡ 3 ≡ 5 ≡ ... mod 2.
Therefore, anywhere we wrote "0" we could have written any other even number,
and similarly "1" could have been replaced by any odd number. For example,
instead of writing 0 × 1 + 1 × 0 ≡ 0 mod 2, an equally valid statement would
have been
10 × (–13) + 27 × 6 ≡ –122 mod 2.
Let's now look at other values for the modulus m. For example, let
m = 3. All multiples of 3 are congruent to each other modulo 3 since the
difference of any two is divisible by 3. Similarly, all numbers of the form
3n + 1 are congruent to each other, and all numbers of the form 3n
+ 2 are congruent to each other.
How about when m = 1? The difference of any two integers is
divisible by 1, so all integers are congruent to each other modulo 1:
... ≡ –3 ≡ –2 ≡ –1 ≡ 0 ≡ 1 ≡ 2 ≡ 3 ≡ ... mod 1.
For this reason, m = 1 is not very interesting, and reducing an
equation modulo 1 doesn't give any information about its solutions.
The modulus m = 12 comes up quite frequently in everyday life, and
its application illustrates a good way to think about modular arithmetic — the
"clock arithmetic" analogy. If it's 7:00, what time will it be in 25 hours?
Since 25 ≡ 1 mod 12, we simply add 1 to 7:
7 + 25 ≡ 7 + 1 ≡ 8 mod 12.
So the clock will read 8:00. Of course, we don't need the formality of
modular arithmetic in order to compute this, but when we do this kind of
computation in our heads, this is really what we are doing.
With m = 12, there are only 12 numbers ("hours") we ever need to think
about. We count them 1, 2, 3, ..., 10, 11, 12, 1, 2, ... , starting over after
12. The numbers 1, 2, ..., 12 represent the twelve equivalence classes modulo
12: Every integer is congruent to exactly one of the numbers 1, 2, ..., 12, just
as the hour on the clock always reads exactly one of 1, 2, ..., 12. These
classes are given by
12n + 1, 12n + 2, 12n + 3, ..., 12n + 11,
12n
as n ranges over the integers.
Of course, the minutes and seconds on a clock are also modular. In these
cases the modulus is m = 60. If we think of the days of the week as
labeled by the numbers 0, 1, 2, 3, 4, 5, 6, then the modulus is m = 7.
The point is that we measure many things, both in mathematics and in real life,
in periodicity, and this can usually be thought of as an application of modular
arithmetic.
III. Properties of Congruence
It is fairly easy to show that for any integers a, b, c,
and m ≠ 0, the following properties hold:
reflexivity: a ≡ a mod m. symmetry: If
a ≡ b mod m, then b ≡ a mod m.
transitivity: If a ≡ b mod m and b ≡
c mod m, then a ≡ c mod m.
Therefore congruence modulo m is an equivalence relation, and this
relation partitions the integers into m equivalence classes:
mn + 0, mn + 1, mn + 2, ..., mn + (m
– 1).
(Since 0 ≡ m mod m, we can either choose 0 or m as a
representative for the first class. It is conventional to choose 0.) To be a
little more formal than we were above, we can write the equivalence class of an
integer a as [a]. The brackets signify that this is an equivalence
class and not simply a number. In this way we can write
(Notice that each congruence symbol "≡" has been replaced by equality "=".)
With this notation, we have the following property for any integers a and
b, which justifies "reduction under addition":
[a + b] = [a] + [b].
Similarly, for any integers a and b we have
[a · b] = [a] · [b],
justifying "reduction under multiplication". (In the fancy language of
abstract algebra, these two properties can be summarized by saying that
reduction modulo m is a homomorphism of rings. In this case the two rings
are the ring of integers and the ring of integers modulo m.)
IV. Exercises
1. Show that the system of equations
11x – 5y = 7, +9x +
10y = –3.
has no integer solutions.
2. Show that the system of equations
24x – 5y = 10, 11x – 9y = 13.
has no integer solutions.
3. Show that if x, y, z are integers such that
x2 + y2 = z2, then at
least one of them is divisible by 2, at least one is divisible by 3, and at
least one is divisible by 5.
4. Show that if x, y, z are integers such that
x3 + y3 = z3, then at
least one of them is divisible by 7.
In
this section all letters stand for integers.
The greatest common divisor of n
and m is noted as (n,m).
Let d = (n,m). All the linear combinations r.n + s.m
of n and m are multiples of d.
If a is a divisor of b we write a | b.
ax = b mod m has a solution x
<=> there is a k and an x such that ax - b = km
<=> there is a k and an x such that ax - km = b
<=> there is a linear combination of a and m that gives b.
<=> b is a multiple of (a,m)
The last assertion holds because all linear combinations of a and m are
exactly all the multiples of (a,m)
To solve the equation ax = b mod m, first check if the
solvability condition is satisfied.
Then d = (a,m) divides a, b and m. So, the equation can be simplified to
ax/d = b/d mod m/d
<=> a'x = b' mod m' with (a',m')=1
If (a',b') is k, then we can use the cancellation law, to simplify the
last equation.
Example:
18x = 6 mod 8
(18,8)=2 is a divisor of 6. The solvability condition is satisfied.
Simplification law gives 9x = 3 mod 4 and the cancellation law gives
3x = 1 mod 4
We start with the simplified equation ax = b mod m with
(a,m)=1.
Take the (m-1) multiples a ; 2a ; 3a ; ... ; (m-1)a all taken mod m .
All these multiples are different!
If for instance 3a = 5a mod m then
canceling a we have 3 = 5 mod m and this is impossible because 3 and 5 are
smaller than m.
Hence, all these multiples are the numbers 1,2,3,...,m-1 in an arbitrary
order.
Therefore, the equation ax = b mod m has just one solution modulo m.
In previous example, we try x = 1, 2 and 3 and we find that (3 mod 4) is the
solution.
Proof:
From above we know that since (a,p)= 1,
the (p-1) multiples
a mod p ; 2a mod p ; 3a mod p ; ... ; (p-1)a mod p
are
the numbers 1,2,3,...,p-1 in an arbitrary order.
Hence,
(1a)(2a)(3a)...((p-1)a)= 1.2.3. ... .(p-1) mod p
Since
(1,p)=(2,p)=(3,p)=..(p-1,p)=1 , we can use the cancellation law.
Proof:
Take all the numbers b from the set {1, 2, ... m} such that
(b,m)=1.
Let b1, b2,..., bn be these
numbers. n = phi(m) .
Take the multiples a.b1 mod m, a.b2 mod m,
...,a.bn mod m
All these numbers are different!
If for instance a.bi =
a.bj mod m then from the cancellation law, we have bi =
bj mod m and this is impossible.
Additionally, each number a.bi is relative prime to m since a and
bi are relative prime to m.
Therefore, these n different numbers
a.b1 mod m,
a.b2 mod m, ...,a.bn mod m
are the n different numbers
relative prime to m.
So, these numbers are b1, b2,...
bn in an arbitrary order.
Hence, (a.b1)(a.b2)...(a.bn) =
b1.b2. ... .bn mod m
Using the cancellation law, we have
aphi(m) = 1 mod m
We start with the simplified equation ax = b mod m with
(a,m)=1.
Then,
a. x = b mod m
multiply both sides with aphi(m)-1
<=> a.aphi(m)-1 . x = b.aphi(m)-1 mod m
<=> x = b.aphi(m)-1 mod m
b.aphi(m)-1 mod m is the unique solution!
Send all suggestions, remarks and reports on errors to
Johan.Claeys@ping.be The subject of the mail must contain the flemish word
'wiskunde' because other mails are filtered to Trash
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