Modular arithmetic/Introduction
Modular arithmetic is a special type of arithmetic that involves only integers.
Contents
The Basic Idea
Let's use a clock as an example, except let's replace the at the top of the clock with a
. Starting at noon, the hour hand points in order to the following:
This is the way in which we count in modulo 12. When we add to
, we arrive back at
. The same is true in any other modulus (modular arithmetic system). In modulo
, we count
We can also count backwards in modulo 5. Any time we subtract 1 from 0, we get 4. So, the integers from to
, when written in modulo 5, are
where is the same as
in modulo 5. Because all integers can be expressed as
,
,
,
, or
in modulo 5, we give these give integers their own name: the modulo 5 residues.
While this may not seem all that useful at first, counting in this way can help us solve an enormous array of number theory problems much more easily!
Congruence
While modulo 5 uses only 5 integers (0 through 4 inclusive), all other integers are said to be congruent to (or the same as) one of those 5 integers in modulo 5. We see how that works by writing the integers going both up and down from 0, then writing the integers modulo 5 below them, making sure that we following the same counting procedure as we would with the hour hand ticking around the clock.
Given integers ,
, and
, with
, we say that
is congruent to
modulo
, or
(mod
), if the difference
is divisible by
.
For a given positive integer , the relation
(mod
) is an equivalence relation on the set of integers. This relation gives rise to an algebraic structure called the integers modulo
(usually known as "the integers mod
," or
for short). This structure gives us a useful tool for solving a wide range of number-theoretic problems, including finding solutions to Diophantine equations, testing whether certain large numbers are prime, and even some problems in cryptology.
Arithmetic Modulo n
Useful Facts
Consider four integers and a positive integer
such that
and
. In modular arithmetic, the following identities hold:
- Addition:
.
- Subtraction:
.
- Multiplication:
.
- Division:
, where
is a positive integer that divides
and
.
- Exponentiation:
where
is a positive integer.
Examples
The Integers Modulo n
The relation (mod
) allows us to divide the set of integers into sets of equivalent elements. For example, if
, then the integers are divided into the following sets:
Notice that if we pick two numbers and
from the same set, then
and
differ by a multiple of
, and therefore
(mod
).
We sometimes refer to one of the sets above by choosing an element from the set, and putting a bar over it. For example, the symbol refers to the set containing
; that is, the set of all integer multiples of
. The symbol
refers to the second set listed above, and
the third. The symbol
refers to the same set as
, and so on.
Instead of thinking of the objects ,
, and
as sets, we can treat them as algebraic objects -- like numbers -- with their own operations of addition and multiplication. Together, these objects form the integers modulo
, or
. More generally, if
is a positive integer, then we can define
,
where for each ,
is defined by
Addition, Subtraction, and Multiplication Mod n
We define addition, subtraction, and multiplication in according to the following rules:
for all
. (Addition)
for all
. (Subtraction)
for all
. (Multiplication)
So for example, if , then we have
Notice that, in each case, we reduce to an answer of the form , where
. We do this for two reasons: to keep possible future calculations as manageable as possible, and to emphasize the point that each expression takes one of only seven (or in general,
) possible values. (Some people find it useful to reduce an answer such as
to
, which is negative but has a smaller absolute value.)
The Natural Appeal of Modular Arithmetic
Observe that we use modular arithmetic even when solving some of the most basic, everyday problems. For example:
Cody is cramming for an exam that will be held at 2 PM. It is the morning of the day of the exam, and Cody did not get any sleep during the night. He knows that it will take him exactly one hour to get to school from the time he wakes up, and he insists upon getting at least five hours of sleep. At what time in the morning should Cody stop studying and go to sleep?
We know that the hours of the day are numbered from to
, with hours having the same number if and only if they are a multiple of
hours apart. So we can use subtraction mod
to answer this question.
We know that since Cody needs five hours of sleep plus one hour to get to school, he must stop studying six hours before the exam. We can find out what time this is by performing the subtraction
So Cody must quit studying at 8 AM.
Of course, we are able to perform calculations like this routinely without a formal understanding of modular arithmetic. One reason for this is that the way we keep time gives us a natural model for addition and subtraction in : a "number circle." Just as we model addition and subtraction by moving along a number line, we can model addition and subtraction mod
by moving along the circumference of a circle. Even though most of us never learn about modular arithmetic in school, we master this computational model at a very early age.
A Word of Caution
Because of the way we define operations in , it is important to check that these operations are well-defined. This is because each of the sets that make up
contains many different numbers, and therefore has many different names. For example, observe that in
, we have
and
. It is reasonable to expect that if we perform the addition
, we should get the same answer as if we compute
, since we are simply using different names for the same objects. Indeed, the first addition yields the sum
, which is the same as the result of the second addition.
The "Useful Facts" above are the key to understanding why our operations yield the same results even when we use different names for the same sets. The task of checking that an operation or function is well-defined, is one of the most important basic techniques in abstract algebra.
Computation of Powers Mod n
The "exponentiation" property given above allows us to perform rapid calculations modulo . Consider, for example, the problem
What are the tens and units digits of ?
We could (in theory) solve this problem by trying to compute , but this would be extremely time-consuming. Moreover, it would give us much more information than we need. Since we want only the tens and units digits of the number in question, it suffices to find the remainder when the number is divided by
. In other words, all of the information we need can be found using arithmetic mod
.
We begin by writing down the first few powers of :
A pattern emerges! We see that (mod
). So for any positive integer
, we have
(mod
). In particular, we can write
(mod
).
By the "multiplication" property above, then, it follows that
(mod
).
Therefore, by the definition of congruence, differs from
by a multiple of
. Since both integers are positive, this means that they share the same tens and units digits. Those digits are
and
, respectively.
A General Algorithm
In the example above, we were fortunate to find a power of -- namely,
-- that is congruent to
mod
. What if we aren't this lucky? Suppose we want to solve the following problem:
What are the tens and units digits of ?
Again, we will solve this problem by computing modulo
. The first few powers of
are
This time, no pattern jumps out at us. Is there a way we can find the power of
without taking this list all the way out to the
term -- or even without patiently waiting for the list to yield a pattern?
Suppose we condense the list we started above; and instead of writing down all powers of , we write only the powers
, where
is a power of
. We have
(Observe that this process yields a pattern of its own, if we carry it out far enough!)
Now, observe that, like any positive integer, can be written as a sum of powers of two:
We can now use this powers-of-two expansion to compute :
So the tens and units digits of are
and
, respectively.
We can use this method to compute modulo
, for any integers
and
, with
. The beauty of this algorithm is that the process takes, at most, approximately
steps -- at most
steps to compute the values
for
a power of two less than
, and at most
steps to multiply the appropriate powers of
according to the binary representation of
.
This method can be further refined using Euler's Totient Theorem.
Applications of Modular Arithmetic
Modular arithmetic is an extremely flexible problem solving tool. The following topics are just a few applications and extensions of its use:
Resources
- The AoPS Introduction to Number Theory by Mathew Crawford.