Difference between revisions of "Modular arithmetic"
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== Intermediate == | == Intermediate == | ||
+ | |||
+ | === Divisibility Tests === | ||
+ | |||
+ | Modular arithmetic is the basis for several well-known divisibility tests. Consider, for example, the test for divisibility by <math>9</math>: | ||
+ | |||
+ | ''Let <math>N</math> be a positive integer. Then <math>N</math> is divisible by <math>9</math> if and only if the sum of the base-ten digits of <math>N</math> is divisible by <math>9</math>.'' | ||
+ | |||
+ | Arithmetic mod <math>9</math> can be used to give an easy proof of this criterion: | ||
+ | |||
+ | Suppose that the base-ten representation of <math>N</math> is | ||
+ | |||
+ | <math>N = a_k a_{k-1} \cdots a_2 a_1 a_0</math>, | ||
+ | |||
+ | where <math>a_i</math> is a digit for each <math>i</math>. Then the numerical value of <math>N</math> is given by | ||
+ | |||
+ | <math>N = a_k \cdot 10^k + a_{k-1} \cdot 10^{k-1} + \cdots + a_1 \cdot 10^1 + a_0 \cdot 10^0</math>. | ||
+ | |||
+ | Now we know that, since <math>10 - 1 = 9</math>, we have <math>10 \equiv 1</math> (mod <math>9</math>). So by the "exponentiation" property above, we have <math>10^j \equiv 1^j \equiv 1</math> (mod <math>9</math>) for every <math>j</math>. | ||
+ | |||
+ | Therefore, by repeated uses of the "addition" and "multiplication" properties, we can write | ||
+ | |||
+ | <math>a_k \cdot 10^k + a_{k-1} \cdot 10^{k-1} + \cdots + a_1 \cdot 10^1 + a_0 \cdot 10^0 \equiv a_k \cdot 1 + a_{k-1} \cdot 1 + \cdots + a_1 \cdot 1 + a_0 \cdot 1</math> (mod <math>9</math>). | ||
+ | |||
+ | Therefore, we have | ||
+ | |||
+ | <math>N \equiv a_k + a_{k-1} + \cdots + a_1 + a_0</math> (mod <math>9</math>). | ||
+ | |||
+ | That is, <math>N</math> differs from the sum of its digits by a multiple of <math>9</math>. It follows, then, that <math>N</math> is a multiple of <math>9</math> if and only if the sum of its digits is a multiple of <math>9</math>. | ||
+ | |||
+ | A virtually identical argument shows that a positive integer is divisible by <math>3</math> if and only if the sum of its base-ten digits is divisible by <math>3</math>. We can also use the technique shown above to devise a test for divisibility by <math>11</math>. | ||
+ | |||
=== Topics === | === Topics === | ||
* [[Fermat's Little Theorem]] | * [[Fermat's Little Theorem]] |
Revision as of 21:17, 24 June 2006
Modular arithmetic is a special type of arithmetic that involves only integers. 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.
Contents
Introductory
Useful Facts
Consider four integers and a positive integer
such that
and
. In modular arithmetic, the following identities hold:
- Addition:
.
- Substraction:
.
- Multiplication:
.
- Division:
, where
is a positive integer that divides
and
.
- Exponentiation:
where
is a positive integer.
Examples
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
.
By writing down the first few powers of , 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.
Applications
Modular arithmetic is an extremely useful tool in mathematics competitions. It enables us to easily solve Linear Diophantine equations, and it often helps with other Diophantine equations as well.
Intermediate
Divisibility Tests
Modular arithmetic is the basis for several well-known divisibility tests. Consider, for example, the test for divisibility by :
Let be a positive integer. Then
is divisible by
if and only if the sum of the base-ten digits of
is divisible by
.
Arithmetic mod can be used to give an easy proof of this criterion:
Suppose that the base-ten representation of is
,
where is a digit for each
. Then the numerical value of
is given by
.
Now we know that, since , we have
(mod
). So by the "exponentiation" property above, we have
(mod
) for every
.
Therefore, by repeated uses of the "addition" and "multiplication" properties, we can write
(mod
).
Therefore, we have
(mod
).
That is, differs from the sum of its digits by a multiple of
. It follows, then, that
is a multiple of
if and only if the sum of its digits is a multiple of
.
A virtually identical argument shows that a positive integer is divisible by if and only if the sum of its base-ten digits is divisible by
. We can also use the technique shown above to devise a test for divisibility by
.
Topics
See also
Miscellany
The binary operation "mod"
Related to the concept of congruence mod is the binary operation
mod
, which is used often in computer programming.
Recall that, by the Division Algorithm, given any two integers and
, with
, we can find integers
and
, with
, such that
. The number
is called the quotient, and the number
is called the remainder. The operation
mod
returns the value of the remainder
. For example:
mod
, since
.
mod
, since
.
mod
, since
.
Observe that if mod
, then we also have
(mod
).