Difference between revisions of "Modular arithmetic/Introduction"

Latest revision as of 07:56, 10 July 2024

Modular arithmetic is a special type of arithmetic that involves only integers. This goal of this article is to explain the basics of modular arithmetic while presenting a progression of more difficult and more interesting problems that are easily solved using modular arithmetic.

Introductory Video

https://youtu.be/7an5wU9Q5hk?t=777

Understand Modular Arithmetic

Let's use a clock as an example, except let's replace the $12$ at the top of the clock with a $0$ .

$[asy]picture pic; path a; a = circle((0,0), 100); draw (a); draw((0,0), linewidth(4)); pair b; b = (0,100); for (int i = 0; i < 12; ++i) { label (pic, (string) i, b); b = rotate(-30,(0,0)) * b; } pic = scale(0.8) * pic; add(pic); draw ((0,0) -- 50*dir(90)); draw ((0,0) -- 70*dir(90));[/asy]$

Starting at noon, the hour hand points in order to the following:

$1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, \ldots$

This is the way in which we count in modulo 12. When we add $1$ to $11$ , we arrive back at $0$ . The same is true in any other modulus (modular arithmetic system). In modulo $5$ , we count

$0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 1, \ldots$

We can also count backwards in modulo 5. Any time we subtract 1 from 0, we get 4. So, the integers from $-12$ to $0$ , when written in modulo 5, are

$3, 4, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0,$

where $-12$ is the same as $3$ in modulo 5. Because all integers can be expressed as $0$ , $1$ , $2$ , $3$ , or $4$ in modulo 5, we give these integers their own name: the residue classes modulo 5. In general, for a natural number $n$ that is greater than 1, the modulo $n$ residues are the integers that are whole numbers less than $n$ :

$0, 1, 2, \ldots, n-1.$

This just relates each integer to its remainder from the Division Theorem. 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!

Residue

We say that $a$ is the modulo- $m$ residue of $n$ when $n\equiv a\pmod m$ , and $0\le a<m$ .

Congruence

There is a mathematical way of saying that all of the integers are the same as one of the modulo 5 residues. For instance, we say that 7 and 2 are congruent modulo 5. We write this using the symbol $\equiv$ : In other words, this means in base 5, these integers have the same residue modulo 5:

$2\equiv 7\equiv 12 \pmod{5}.$

The (mod 5) part just tells us that we are working with the integers modulo 5. In modulo 5, two integers are congruent when their difference is a multiple of 5. In general, two integers $a$ and $b$ are congruent modulo $n$ when $a - b$ is a multiple of $n$ . In other words, $a \equiv b \pmod{n}$ when $\frac{a-b}{n}$ is an integer. Otherwise, $a \not\equiv b \pmod{n}$ , which means that $a$ and $b$ are not congruent modulo $n$ .

Examples

$31 \equiv 1 \pmod{10}$ because $31 - 1 = 30$ is a multiple of $10$ .

$43 \equiv 22 \pmod{7}$ because $\frac{43 - 22}{7} = \frac{21}{7} = 3$ , which is an integer.

$8 \not\equiv -8 \pmod{3}$ because $8 - (-8) = 16$ , which is not a multiple of $3$ .

$91 \not\equiv 18 \pmod{6}$ because $\frac{91 - 18}{6} = \frac{73}{6}$ , which is not an integer.

Sample Problem

Find the modulo $4$ residue of $311$ .

Solution:

Since $311 \div 4 = 77$ R $3$ , we know that

$311 \equiv 3 \pmod{4}$

and $3$ is the modulo $4$ residue of $311$ .

Another Solution:

Since $311 = 300 + 11$ , we know that

$311 \equiv 0+11 \pmod{4}$

We can now solve it easily

$11 \equiv 3 \pmod{4}$

and $3$ is the modulo $4$ residue of $311$

Another Solution:

We know $308$ is a multiple of $4$ since $8$ is a multiple of $4$ . Thus, $311-308=3$ and $3$ is the modulo $4$ residue of $311$ .

Making Computation Easier

We don't always need to perform tedious computations to discover solutions to interesting problems. If all we need to know about are remainders when integers are divided by $n$ , then we can work directly with those remainders in modulo $n$ . This can be more easily understood with a few examples.

Addition

Problem

Suppose we want to find the units digit of the following sum:

$2403 + 791 + 688 + 4339.$

We could find their sum, which is $8221$ , and note that the units digit is $1$ . However, we could find the units digit with far less calculation.

Solution

We can simply add the units digits of the addends:

$3 + 1 + 8 + 9 = 21.$

The units digit of this sum is $1$ , which must be the same as the units digit of the four-digit sum we computed earlier.

Why we only need to use remainders

We can rewrite each of the integers in terms of multiples of $10$ and remainders:
$2403 = 240 \cdot 10 + 3$
$791 = 79 \cdot 10 + 1$
$688 = 68 \cdot 10 + 8$
$4339 = 433 \cdot 10 + 9$ .
When we add all four integers, we get

$(240 \cdot 10 + 3) + (79 \cdot 10 + 1) + (68 \cdot 10 + 8) + (433 \cdot 10 + 9)$ $= (240 + 79 + 68 + 433) \cdot 10 + (3 + 1 + 8 + 9)$

At this point, we already see the units digits grouped apart and added to a multiple of $10$ (which will not affect the units digit of the sum):

$= 820 \cdot 10 + 21 = 8200 + 21 = 8221$ .

Solution using modular arithmetic

Now let's look back at this solution, using modular arithmetic from the start. Note that
$2403 \equiv 3 \pmod{10}$
$791 \equiv 1 \pmod{10}$
$688 \equiv 8 \pmod{10}$
$4339 \equiv 9 \pmod{10}$
Because we only need the modulo $10$ residue of the sum, we add just the residues of the summands:

$2403 + 791 + 688 + 4339 \equiv 3 + 1 + 8 + 9 \equiv 21 \equiv 1 \pmod{10},$

so the units digit of the sum is just $1$ .

Addition rule

In general, when $a, b, c$ , and $d$ are integers and $m$ is a positive integer such that

$a \equiv c \pmod{m}$ $b \equiv d \pmod{m}$

the following is always true:

$a + b \equiv c + d \pmod{m}$ .

And as we did in the problem above, we can apply more pairs of equivalent integers to both sides, just repeating this simple principle.

Proof of the addition rule

Let $a-c=m\cdot k$ , and $b-d=m\cdot l$ where $l$ and $k$ are integers. Adding the two equations we get: $\begin{eqnarray*} mk+ml&=&(a-c)+(b-d)\\ m(k+l)&=&(a+b)-(c+d) \end{eqnarray*}$

Which is equivalent to saying $a+b\equiv c+d\pmod{m}$

Subtraction

The same shortcut that works with addition of remainders works also with subtraction.

Problem

Find the remainder when the difference between $60002$ and $601$ is divided by $6$ .

Solution

Note that $60002 = 10000 \cdot 6 + 2$ and $601 = 100 \cdot 6 + 1$ . So,
$60002 \equiv 2 \pmod{6}$
$601 \equiv 1 \pmod{6}$
Thus,

$60002 - 601 \equiv 2 - 1 \equiv 1 \pmod{6},$

so 1 is the remainder when the difference is divided by $6$ . (Perform the subtraction yourself, divide by $6$ , and see!)

Subtraction rule

When $a, b, c$ , and $d$ are integers and $m$ is a positive integer such that

$a \equiv c \pmod{m}$ $b \equiv d \pmod{m}$

the following is always true:

$a - b \equiv c - d \pmod{m} .$

Multiplication

Modular arithmetic provides an even larger advantage when multiplying than when adding or subtracting. Let's take a look at a problem that demonstrates the point.

Problem

Jerry has $44$ boxes of soda in his truck. The cans of soda in each box are packed oddly so that there are $113$ cans of soda in each box. Jerry plans to pack the sodas into cases of $12$ cans to sell. After making as many complete cases as possible, how many sodas will Jerry have leftover?

Solution

First, we note that this word problem is asking us to find the remainder when the product $44 \cdot 113$ is divided by $12$ .

Now, we can write each $44$ and $113$ in terms of multiples of $12$ and remainders:
$44 = 3 \cdot 12 + 8$
$113 = 9 \cdot 12 + 5$
This gives us a nice way to view their product:

$44 \cdot 113 = (3 \cdot 12 + 8)(9 \cdot 12 + 5)$ Using FOIL, we get that this equals $(3 \cdot 9) \cdot 12^2 + (3 \cdot 5 + 8 \cdot 9) \cdot 12 + (8 \cdot 5)$

We can already see that each part of the product is a multiple of $12$ , except the product of the remainders when each $44$ and $113$ are divided by 12. That part of the product is $8 \cdot 5 = 40$ , which leaves a remainder of $4$ when divided by $12$ . So, Jerry has $4$ sodas leftover after making as many cases of $12$ as possible.

Solution using modular arithmetic

First, we note that
$44 \equiv 8 \pmod{12}$
$113 \equiv 5 \pmod{12}$
Thus,

$44 \cdot 113 \equiv 8 \cdot 5 \equiv 40 \equiv 4 \pmod{12},$

meaning there are $4$ sodas leftover. Yeah, that was much easier.

Multiplication rule

When $a, b, c$ , and $d$ are integers and $m$ is a positive integer such that

$a \equiv c \pmod{m}$ $b \equiv d \pmod{m}$

The following is always true:

$a \cdot b \equiv c \cdot d \pmod{m}$ .

Exponentiation

Since exponentiation is just repeated multiplication, it makes sense that modular arithmetic would make many problems involving exponents easier. In fact, the advantage in computation is even larger and we explore it a great deal more in the intermediate modular arithmetic article.

Note to everybody: Exponentiation is very useful as in the following problem:

Problem #1

What is the last digit of $(...((7)^7)^7)...)^7$ if there are 1000 7s as exponents and only one 7 in the middle?

We can solve this problem using mods. This can also be stated as $7^{7^{1000}}$ . After that, we see that 7 is congruent to -1 in mod 4, so we can use this fact to replace the 7s with -1s, because 7 has a pattern of repetitive period 4 for the units digit. $(1)^{1000}$ is simply 1, so therefore $7^1=7$ , which really is the last digit.

Problem #2

What are the tens and units digits of $7^{1942}$ ?

We could (in theory) solve this problem by trying to compute $7^{1942}$ , 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 $100$ . In other words, all of the information we need can be found using arithmetic mod $100$ .

We begin by writing down the first few powers of $7$ mod $100$ :

$7, 49, 43, 1, 7, 49, 43, 1, \ldots$

A pattern emerges! We see that $7^4 = 2401 \equiv 1$ (mod $100$ ). So for any positive integer $k$ , we have $7^{4k} = (7^4)^k \equiv 1^k \equiv 1$ (mod $100$ ). In particular, we can write

$7^{1940} = 7^{4 \cdot 485} \equiv 1$ (mod $100$ ).

By the "multiplication" property above, then, it follows that

$7^{1942} = 7^{1940} \cdot 7^2 \equiv 1 \cdot 7^2 \equiv 49$ (mod $100$ ).

Therefore, by the definition of congruence, $7^{1942}$ differs from $49$ by a multiple of $100$ . Since both integers are positive, this means that they share the same tens and units digits. Those digits are $4$ and $9$ , respectively.

Problem #3

Can you find a number that is both a multiple of $2$ but not a multiple of $4$ and a perfect square?

No, you cannot. Rewriting the question, we see that it asks us to find an integer $n$ that satisfies $4n+2=x^2$ .

Taking mod $4$ on both sides, we find that $x^2\equiv 2\pmod{4}$ . Now, all we are missing is proof that no matter what $x$ is, $x^2$ will never be a multiple of $4$ plus $2$ , so we work with cases:

$x\equiv 0\pmod{4}\implies x^2\equiv 0\pmod{4}$

$x\equiv 1\pmod{4}\implies x^2\equiv 1\pmod{4}$

$x\equiv 2\pmod{4}\implies x^2\equiv 4\equiv 0\pmod{4}$

$x\equiv 3\pmod{4}\implies x^2\equiv 9\equiv 1\pmod{4}$

This assures us that it is impossible to find such a number.

Summary of Useful Facts

Consider four integers ${a},{b},{c},{d}$ and a positive integer ${m}$ such that $a\equiv b\pmod {m}$ and $c\equiv d\pmod {m}$ . In modular arithmetic, the following identities hold:

Addition: $a+c\equiv b+d\pmod {m}$ .
Subtraction: $a-c\equiv b-d\pmod {m}$ .
Multiplication: $ac\equiv bd\pmod {m}$ .
Division: $\frac{a}{e}\equiv \frac{b}{e}\pmod {\frac{m}{\gcd(m,e)}}$ , where $e$ is a positive integer that divides ${a}$ and $b$ .
Exponentiation: $a^e\equiv b^e\pmod {m}$ where $e$ is a positive integer.

Problem Applications

https://artofproblemsolving.com/wiki/index.php/2011_AMC_8_Problems/Problem_22

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.
The AoPS Introduction to Number Theory Course. Thousands of students have learned more about modular arithmetic and problem solving from this 12 week class.

@@ Line 1: / Line 1: @@
-'''Modular arithmetic''' is a special type of arithmetic that involves only [[integers]].
+'''[[Modular arithmetic]]''' is a special type of arithmetic that involves only [[integers]].  This goal of this article is to explain the basics of modular arithmetic while presenting a progression of more difficult and more interesting problems that are easily solved using modular arithmetic.
-== The Basic Idea ==
+==Introductory Video==
-Let's use a clock as an example, except let's replace the <math>\displaystyle 12</math> at the top of the clock with a <math>\displaystyle 0</math>.  Starting at noon, the hour hand points in order to the following:
+https://youtu.be/7an5wU9Q5hk?t=777
+==Understand Modular Arithmetic==
+Let's use a clock as an example, except let's replace the <math>12</math> at the top of the clock with a <math>0</math>.
-<math>\displaystyle 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, \ldots </math>
+<asy>picture pic;
+path a;
+a = circle((0,0), 100);
+draw (a);
+draw((0,0), linewidth(4));
+pair b;
+b = (0,100);
-This is the way in which we count in '''modulo 12'''.  When we add <math>\displaystyle 1</math> to <math>\displaystyle 11</math>, we arrive back at <math>\displaystyle 0</math>.  The same is true in any other [[modulus]] (modular arithmetic system).  In modulo <math>\displaystyle 5</math>, we [[counting | count]]
+for (int i = 0; i < 12; ++i)
+{
+label  (pic, (string) i, b);
+b = rotate(-30,(0,0)) * b;
+}
+pic = scale(0.8) * pic;
+add(pic);
+draw ((0,0) -- 50*dir(90));
+draw ((0,0) -- 70*dir(90));</asy>
-<math>\displaystyle 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 1, \ldots</math>
+Starting at noon, the hour hand points in order to the following:
+<math>1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, \ldots </math>
-We can also count backwards in modulo 5.  Any time we subtract 1 from 0, we get 4.  So, the integers from <math>\displaystyle -12</math> to <math>\displaystyle 0</math>, when written in modulo 5, are
+This is the way in which we count in '''modulo 12'''.  When we add <math>1</math> to <math>11</math>, we arrive back at <math>0</math>.  The same is true in any other [[modulus]] (modular arithmetic system).  In modulo <math>5</math>, we [[counting | count]]
-<math>\displaystyle 3, 4, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0,</math>
+<math>0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 1, \ldots</math>
-where <math>\displaystyle -12</math> is the same as <math>\displaystyle 3</math> in modulo 5.  Because all integers can be expressed as <math>\displaystyle 0</math>, <math>\displaystyle 1</math>, <math>\displaystyle 2</math>, <math>\displaystyle 3</math>, or <math>\displaystyle 4</math> in modulo 5, we give these give integers their own name: the modulo 5 '''[[residue | residues]]'''.  In general, for a natural number <math>\displaystyle n</math> that is greater than 1, the modulo <math>\displaystyle n</math> residues are the integers that are [[whole number | whole numbers]] less than <math>\displaystyle n</math>:
+We can also count backwards in modulo 5.  Any time we subtract 1 from 0, we get 4.  So, the integers from <math>-12</math> to <math>0</math>, when written in modulo 5, are
-<math>\displaystyle 0, 1, 2, \ldots, n-1. </math>
+<math>3, 4, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0,</math>
+where <math>-12</math> is the same as <math>3</math> in modulo 5.  Because all integers can be expressed as <math>0</math>, <math>1</math>, <math>2</math>, <math>3</math>, or <math>4</math> in modulo 5, we give these integers their own name: the '''[[residue class]]es''' modulo 5.  In general, for a natural number <math>n</math> that is greater than 1, the modulo <math>n</math> residues are the integers that are [[whole number | whole numbers]] less than <math>n</math>:
+<math>0, 1, 2, \ldots, n-1. </math>
 This just relates each integer to its [[remainder]] from the [[Division Theorem]].  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!
+==Residue==
+We say that <math>a</math> is the modulo-<math>m</math> <b>residue</b> of <math>n</math> when <math>n\equiv a\pmod m</math>, and <math>0\le a<m</math>.
 == Congruence ==
-There is a mathematical way of saying that all of the integers are the same as one of the modulo 5 residues.  For instance, we say that 7 and 2 are '''[[congruent]]''' modulo 5.  We write this using the symbol <math>\displaystyle \equiv</math>:
+There is a mathematical way of saying that all of the integers are the same as one of the modulo 5 residues.  For instance, we say that 7 and 2 are '''[[congruent]]''' modulo 5.  We write this using the symbol <math>\equiv</math>: In other words, this means in base 5, these integers have the same residue modulo 5:
-<center><math>\displaystyle 7 \equiv 2 \pmod{5}. </math></center>
+<center><math>2\equiv 7\equiv 12 \pmod{5}. </math></center>
-The '''(mod 5)''' part just tells us that we are working with the integers modulo 5.  In modulo 5, two integers are congruent when their difference is a [[multiple]] of 5.  Thus each of the following integers is congruent modulo 5:
+The '''(mod 5)''' part just tells us that we are working with the integers modulo 5.  In modulo 5, two integers are congruent when their difference is a [[multiple]] of 5.
-<center><math>\displaystyle -12 \equiv -7 \equiv -2 \equiv 3 \equiv 8 \equiv 13 \equiv 18 \equiv 23 \pmod{5} </math></center>
+In general, two integers <math>a</math> and <math>b</math> are congruent modulo <math>n</math> when <math>a - b</math> is a multiple of <math>n</math>.  In other words, <math>a \equiv b \pmod{n}</math> when <math>\frac{a-b}{n}</math> is an integer.  Otherwise, <math>a \not\equiv b \pmod{n}</math>, which means that <math>a</math> and <math>b</math> are '''not congruent''' modulo <math>n</math>.
-In general, two integers <math>\displaystyle a</math> and <math>\displaystyle b</math> are congruent modulo <math>\displaystyle n</math> when <math>\displaystyle a - b</math> is a multiple of <math>\displaystyle n</math>.  In other words, <math>\displaystyle a \equiv b \pmod{n}</math> when <math>\displaystyle \frac{a-b}{n}</math> is an integer.  Otherwise, <math>\displaystyle a \not\equiv b \pmod{n}</math>, which means that <math>\displaystyle a</math> and <math>\displaystyle b</math> are '''not congruent''' modulo <math>\displaystyle n</math>.
 === Examples ===
-* <math>\displaystyle 31 \equiv 1 \pmod{10}</math> because <math>\displaystyle 31 - 1 = 30</math> is a multiple of <math>\displaystyle 10</math>
+* <math>31 \equiv 1 \pmod{10}</math> because <math>31 - 1 = 30</math> is a multiple of <math>10</math>.
-* <math>\displaystyle 43 \equiv 22 \pmod{7}</math> because <math>\displaystyle \frac{43 - 22}{7} = \frac{21}{7} = 3</math>, which is an integer.
+* <math>43 \equiv 22 \pmod{7}</math> because <math>\frac{43 - 22}{7} = \frac{21}{7} = 3</math>, which is an integer.
-* <math>\displaystyle 8 \not\equiv -8 \pmod{3}</math> because <math>\displaystyle 8 - (-8) = 16</math>, which is not a multiple of <math>\displaystyle 3</math>.
+* <math>8 \not\equiv -8 \pmod{3}</math> because <math>8 - (-8) = 16</math>, which is not a multiple of <math>3</math>.
-* <math>\displaystyle 91 \not\equiv 18 \pmod{6}</math> because <math>\displaystyle \frac{91 - 18}{6} = \frac{73}{6}</math>, which is not an integer.
+* <math>91 \not\equiv 18 \pmod{6}</math> because <math>\frac{91 - 18}{6} = \frac{73}{6}</math>, which is not an integer.
 === Sample Problem ===
-Find the modulo <math>\displaystyle 4</math> residue of <math>\displaystyle 311</math>.
+Find the modulo <math>4</math> residue of <math>311</math>.
 ==== Solution: ====
-Since <math>\displaystyle 311 \div 4 = 77</math> R <math>\displaystyle 3</math>, we know that
+Since <math>311 \div 4 = 77</math> R <math>3</math>, we know that
-<center><math>\displaystyle 311 \equiv 3 \pmod{4}</math></center>
+<center><math>311 \equiv 3 \pmod{4}</math></center>
-and <math>\displaystyle 3</math> is the modulo <math>\displaystyle 4</math> residue of <math>\displaystyle 311</math>.
+and <math>3</math> is the modulo <math>4</math> residue of <math>311</math>.
+==== Another Solution: ====
+Since <math> 311 = 300 + 11 </math>, we know that
+<center><math>311 \equiv 0+11 \pmod{4}</math></center>
+We can now solve it easily
+<center><math>11 \equiv 3 \pmod{4}</math></center>
+and <math>3</math> is the modulo <math>4</math> residue of <math>311</math>
+==== Another Solution: ====
+We know <math>308</math> is a multiple of <math>4</math> since <math>8</math> is a multiple of <math>4</math>. Thus, <math>311-308=3</math> and <math>3</math> is the modulo <math>4</math> residue of <math>311</math>.
 == Making Computation Easier ==
-We don't always need to perform tedious computations to discover solutions to interesting problems.  If all we need to know about are remainders when integers are divided by <math>\displaystyle n</math>, then we can work directly with those remainders in modulo <math>\displaystyle n</math>.  This can be more easily understood with a few examples.
+We don't always need to perform tedious computations to discover solutions to interesting problems.  If all we need to know about are remainders when integers are divided by <math>n</math>, then we can work directly with those remainders in modulo <math>n</math>.  This can be more easily understood with a few examples.
 === Addition ===
@@ Line 76: / Line 107: @@
-<center><math>\displaystyle 2403 + 791 + 688 + 4339.</math></center>
+<center><math>2403 + 791 + 688 + 4339.</math></center>
-We could find their sum, which is <math>\displaystyle 8221</math>, and note that the units digit is <math>\displaystyle 1</math>.  However, we could find the units digit with far less calculation.
+We could find their sum, which is <math>8221</math>, and note that the units digit is <math>1</math>.  However, we could find the units digit with far less calculation.
 ==== Solution ====
-We can simply add the units digits of the [[summand | summands]]:
+We can simply add the units digits of the addends:
-<center><math>\displaystyle 3 + 1 + 8 + 9 = 21.</math></center>
+<center><math>3 + 1 + 8 + 9 = 21.</math></center>
-The units digit of this sum is <math>\displaystyle 1</math>, which ''must'' be the same as the units digit of the four-digit sum we computed earlier.
+The units digit of this sum is <math>1</math>, which ''must'' be the same as the units digit of the four-digit sum we computed earlier.
 ==== Why we only need to use remainders ====
-We can rewrite each of the integers in terms of multiples of <math>\displaystyle 10</math> and remainders:<br>
+We can rewrite each of the integers in terms of multiples of <math>10</math> and remainders:<br>
-<math>\displaystyle 2403 = 240 \cdot 10 + 3</math><br>
+<math>2403 = 240 \cdot 10 + 3</math><br>
-<math>\displaystyle 791 = 79 \cdot 10 + 1</math><br>
+<math>791 = 79 \cdot 10 + 1</math><br>
-<math>\displaystyle 688 = 68 \cdot 10 + 8</math><br>
+<math>688 = 68 \cdot 10 + 8</math><br>
-<math>\displaystyle 4339 = 433 \cdot 10 + 9</math><br>
+<math>4339 = 433 \cdot 10 + 9</math>.<br>
 When we add all four integers, we get
@@ Line 104: / Line 135: @@
-At this point we already see the units digits grouped apart and added to a multiple of <math>\displaystyle 10</math> (which will not affect the units digit of the sum):
+At this point, we already see the units digits grouped apart and added to a multiple of <math>10</math> (which will not affect the units digit of the sum):
+<center><math>= 820 \cdot 10 + 21 = 8200 + 21 = 8221</math>.</center>
+==== Solution using modular arithmetic ====
+Now let's look back at this solution, using modular arithmetic from the start.  Note that<br>
+<math>2403 \equiv 3 \pmod{10}</math><br>
+<math>791 \equiv 1 \pmod{10}</math><br>
+<math>688 \equiv 8 \pmod{10}</math><br>
+<math>4339 \equiv 9 \pmod{10}</math><br>
+Because we only need the modulo <math>10</math> residue of the sum, we add just the residues of the summands:
-<center><math>= 820 \cdot 10 + 21 = 8200 + 21 = 8221</math></center>
+<center><math>2403 + 791 + 688 + 4339 \equiv 3 + 1 + 8 + 9 \equiv 21 \equiv 1 \pmod{10},</math></center>
+so the units digit of the sum is just <math>1</math>.
+==== Addition rule ====
+In general, when <math>a, b, c</math>, and <math>d</math> are integers and <math>m</math> is a positive integer such that<br>
+<center><math>a \equiv c \pmod{m} </math></center>
+<center><math>b \equiv d \pmod{m} </math></center>
+the following is always true:
-== Summary of Useful Facts ==
+<center><math>a + b \equiv c + d \pmod{m} </math>.</center>
+And as we did in the problem above, we can apply more pairs of equivalent integers to both sides, just repeating this simple principle.
-Consider four integers <math>{a},{b},{c},{d}</math> and a positive integer <math>{m}</math> such that <math>a\equiv b\pmod {m}</math> and <math>c\equiv d\pmod {m}</math>. In modular arithmetic, the following [[identity | identities]] hold:
+====Proof of the addition rule====
-* Addition: <math>a+c\equiv b+d\pmod {m}</math>.
+Let <math>a-c=m\cdot k</math>, and <math>b-d=m\cdot l</math> where <math>l</math> and <math>k</math> are integers.
-* Subtraction: <math>a-c\equiv b-d\pmod {m}</math>.
+Adding the two equations we get:
-* Multiplication: <math>ac\equiv bd\pmod {m}</math>.
+<cmath>
-* Division: <math>\frac{a}{e}\equiv \frac{b}{e}\pmod {\frac{m}{\gcd(m,e)}}</math>, where <math>e</math> is a positive integer that divides <math>{a}</math> and <math>b</math>.
+\begin{eqnarray*}
-* Exponentiation: <math>a^e\equiv b^e\pmod {m}</math> where <math>e</math> is a positive integer.
+mk+ml&=&(a-c)+(b-d)\\
+m(k+l)&=&(a+b)-(c+d)
+\end{eqnarray*}
+</cmath>
+Which is equivalent to saying <math>a+b\equiv c+d\pmod{m}</math>
+=== Subtraction ===
+The same shortcut that works with addition of remainders works also with subtraction.
-== The Natural Appeal of Modular Arithmetic ==
+==== Problem ====
+Find the remainder when the difference between <math>60002</math> and <math>601</math> is divided by <math>6</math>.
-Observe that we use modular arithmetic even when solving some of the most basic, everyday problems.  For example:
+==== Solution ====
+Note that <math>60002 = 10000 \cdot 6 + 2</math> and <math>601 = 100 \cdot 6 + 1</math>.  So,<br>
+<math>60002 \equiv 2 \pmod{6}</math><br>
+<math>601 \equiv 1 \pmod{6}</math><br>
+Thus,
+<center><math>60002 - 601 \equiv 2 - 1 \equiv 1 \pmod{6}, </math></center>
+so 1 is the remainder when the difference is divided by <math>6</math>. (Perform the subtraction yourself, divide by <math>6</math>, and see!)
-''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?''
+==== Subtraction rule ====
+When <math>a, b, c</math>, and <math>d</math> are integers and <math>m</math> is a positive integer such that<br>
+<center><math>a \equiv c \pmod{m} </math></center>
+<center><math>b \equiv d \pmod{m} </math></center>
+the following is always true:
-We know that the hours of the day are numbered from <math>1</math> to <math>12</math>, with hours having the same number if and only if they are a multiple of <math>12</math> hours apart.  So we can use subtraction mod <math>12</math> to answer this question.
+<center><math>a - b \equiv c - d \pmod{m} .</math></center>
-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
+=== Multiplication ===
+Modular arithmetic provides an even larger advantage when multiplying than when adding or subtracting.  Let's take a look at a problem that demonstrates the point.
-<math>2 - 6 = -4 \equiv 8 \pmod{12}</math>.
+==== Problem ====
+Jerry has <math>44</math> boxes of soda in his truck.  The cans of soda in each box are packed oddly so that there are <math>113</math> cans of soda in each box.  Jerry plans to pack the sodas into cases of <math>12</math> cans to sell.  After making as many complete cases as possible, how many sodas will Jerry have leftover?
-So Cody must quit studying at 8 AM.
+==== Solution ====
+First, we note that this [[word problem]] is asking us to find the remainder when the product <math>44 \cdot 113</math> is divided by <math>12</math>.
-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 <math>\mathbb{Z}_n</math>:  a "number circle."  Just as we model addition and subtraction by moving along a number line, we can model addition and subtraction mod <math>n</math> 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.
+Now, we can write each <math>44</math> and <math>113</math> in terms of multiples of <math>12</math> and remainders:<br>
+<math>44 = 3 \cdot 12 + 8</math><br>
+<math>113 = 9 \cdot 12 + 5</math><br>
+This gives us a nice way to view their product:
-=== Computation of Powers Mod n ===
-The "exponentiation" property given above allows us to perform rapid calculations modulo <math>n</math>.  Consider, for example, the problem
+<cmath>44 \cdot 113 = (3 \cdot 12 + 8)(9 \cdot 12 + 5)</cmath>
+Using [[FOIL]], we get that this equals
+<cmath>(3 \cdot 9) \cdot 12^2 + (3 \cdot 5 + 8 \cdot 9) \cdot 12 + (8 \cdot 5)</cmath>
-''What are the tens and units digits of <math>7^{1942}</math>?''
-We could (in theory) solve this problem by trying to compute <math>7^{1942}</math>, 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 <math>100</math>.  In other words, all of the information we need can be found using arithmetic mod <math>100</math>.
+We can already see that each part of the product is a multiple of <math>12</math>, except the product of the remainders when each <math>44</math> and <math>113</math> are divided by 12.  That part of the product is <math>8 \cdot 5 = 40</math>, which leaves a remainder of <math>4</math> when divided by <math>12</math>.  So, Jerry has <math>4</math> sodas leftover after making as many cases of <math>12</math> as possible.
-We begin by writing down the first few powers of <math>7</math> mod <math>100</math>:
+==== Solution using modular arithmetic ====
+First, we note that<br>
+<math>44 \equiv 8 \pmod{12}</math><br>
+<math>113 \equiv 5 \pmod{12}</math><br>
+Thus,
+<center><math>44 \cdot 113 \equiv 8 \cdot 5 \equiv 40 \equiv 4 \pmod{12},</math></center>
+meaning there are <math>4</math> sodas leftover.  Yeah, that was much easier.
-<math>7, 49, 43, 1, 7, 49, 43, 1, \ldots</math>
+==== Multiplication rule ====
+When <math>a, b, c</math>, and <math>d</math> are integers and <math>m</math> is a positive integer such that<br>
+<center><math>a \equiv c \pmod{m} </math></center>
+<center><math>b \equiv d \pmod{m} </math></center>
+The following is always true:
-A pattern emerges!  We see that <math>7^4 = 2401 \equiv 1</math> (mod <math>100</math>).  So for any positive integer <math>k</math>, we have <math>7^{4k} = (7^4)^k \equiv 1^k \equiv 1</math> (mod <math>100</math>).  In particular, we can write
+<center><math>a \cdot b \equiv c \cdot d \pmod{m} </math>.</center>
-<math>7^{1940} = 7^{4 \cdot 485} \equiv 1</math> (mod <math>100</math>).
-By the "multiplication" property above, then, it follows that
+=== Exponentiation ===
+Since [[exponentiation]] is just repeated multiplication, it makes sense that modular arithmetic would make many problems involving exponents easier.  In fact, the advantage in computation is even larger and we explore it a great deal more in the [[intermediate modular arithmetic]] article.
-<math>7^{1942} = 7^{1940} \cdot 7^2 \equiv 1 \cdot 7^2 \equiv 49</math> (mod <math>100</math>).
+Note to everybody: Exponentiation is very useful as in the following problem:
-Therefore, by the definition of congruence, <math>7^{1942}</math> differs from <math>49</math> by a multiple of <math>100</math>.  Since both integers are positive, this means that they share the same tens and units digits.  Those digits are <math>4</math> and <math>9</math>, respectively.
+==== Problem #1====
+What is the last digit of <math>(...((7)^7)^7)...)^7</math> if there are 1000 7s as exponents and only one 7 in the middle?
-==== A General Algorithm ====
+We can solve this problem using mods. This can also be stated as <math>7^{7^{1000}}</math>.  After that, we see that 7 is congruent to -1 in mod 4, so we can use this fact to replace the 7s with -1s, because 7 has a pattern of repetitive period 4 for the units digit.  <math>(1)^{1000}</math> is simply 1, so therefore <math>7^1=7</math>, which really is the last digit.
-In the example above, we were fortunate to find a power of <math>7</math> -- namely, <math>7^4</math> -- that is congruent to <math>1</math> mod <math>100</math>.  What if we aren't this lucky?  Suppose we want to solve the following problem:
+==== Problem #2====
+What are the tens and units digits of <math>7^{1942}</math>?
-''What are the tens and units digits of <math>13^{404}</math>?''
+We could (in theory) solve this problem by trying to compute <math>7^{1942}</math>, 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 <math>100</math>.  In other words, all of the information we need can be found using arithmetic mod <math>100</math>.
-Again, we will solve this problem by computing <math>13^{404}</math> modulo <math>100</math>.  The first few powers of <math>13</math> mod <math>100</math> are
+We begin by writing down the first few powers of <math>7</math> mod <math>100</math>:
-<math>13, 69, 97, 61, 93, \ldots</math>
+<math>7, 49, 43, 1, 7, 49, 43, 1, \ldots</math>
-This time, no pattern jumps out at us.  Is there a way we can find the <math>404^{th}</math> power of <math>13</math> mod <math>100</math> without taking this list all the way out to the <math>404^{th}</math> term -- or even without patiently waiting for the list to yield a pattern?
+A pattern emerges!  We see that <math>7^4 = 2401 \equiv 1</math> (mod <math>100</math>).  So for any positive integer <math>k</math>, we have <math>7^{4k} = (7^4)^k \equiv 1^k \equiv 1</math> (mod <math>100</math>).  In particular, we can write
-Suppose we condense the list we started above; and instead of writing down all powers of <math>13</math> mod <math>100</math>, we write only the powers <math>13^k</math>, where <math>k</math> is a power of <math>2</math>.  We have the following (all congruences are modulo <math>100</math>):
+<math>7^{1940} = 7^{4 \cdot 485} \equiv 1</math> (mod <math>100</math>).
-<math>\displaystyle 13^1 = 13</math>
+By the "multiplication" property above, then, it follows that
-<math>13^2 \equiv 69</math>
+<math>7^{1942} = 7^{1940} \cdot 7^2 \equiv 1 \cdot 7^2 \equiv 49</math> (mod <math>100</math>).
-<math>13^4 \equiv 69^2 \equiv 61</math>
+Therefore, by the definition of congruence, <math>7^{1942}</math> differs from <math>49</math> by a multiple of <math>100</math>.  Since both integers are positive, this means that they share the same tens and units digits.  Those digits are <math>4</math> and <math>9</math>, respectively.
-<math>13^8 \equiv 61^2 \equiv 21</math>
+==== Problem #3====
-<math>13^{16} \equiv 21^2 \equiv 41</math>
+Can you find a number that is both a multiple of <math>2</math> but not a multiple of <math>4</math> and a perfect square?
-<math>13^{32} \equiv 41^2 \equiv 81</math>
+No, you cannot. Rewriting the question, we see that it asks us to find an integer <math>n</math> that satisfies <math>4n+2=x^2</math>.
-<math>13^{64} \equiv 81^2 \equiv 61</math>
+Taking mod <math>4</math> on both sides, we find that <math>x^2\equiv 2\pmod{4}</math>. Now, all we are missing is proof that no matter what <math>x</math> is, <math>x^2</math> will never be a multiple of <math>4</math> plus <math>2</math>, so we work with cases:
-<math>13^{128} \equiv 61^2 \equiv 21</math>
+<math>x\equiv 0\pmod{4}\implies x^2\equiv 0\pmod{4}</math>
-<math>13^{256} \equiv 21^2 \equiv 41</math>
+<math>x\equiv 1\pmod{4}\implies x^2\equiv 1\pmod{4}</math>
-(Observe that this process yields a pattern of its own, if we carry it out far enough!)
+<math>x\equiv 2\pmod{4}\implies x^2\equiv 4\equiv 0\pmod{4}</math>
-Now, observe that, like any positive integer, <math>404</math> can be written as a sum of powers of two:
+<math>x\equiv 3\pmod{4}\implies x^2\equiv 9\equiv 1\pmod{4}</math>
-<math>404 = 256 + 128 + 16 + 4</math>
+This assures us that it is impossible to find such a number.
-We can now use this powers-of-two expansion to compute <math>\overline{13}^{404}</math>:
+== Summary of Useful Facts ==
-<math>13^{404} = 13^{256} \cdot 13^{128} \cdot 13^{16} \cdot 13^4 \equiv 41 \cdot 21 \cdot 41 \cdot 61 \equiv 61.</math>
+Consider four integers <math>{a},{b},{c},{d}</math> and a positive integer <math>{m}</math> such that <math>a\equiv b\pmod {m}</math> and <math>c\equiv d\pmod {m}</math>. In modular arithmetic, the following [[identity | identities]] hold:
-So the tens and units digits of <math>13^{404}</math> are <math>6</math> and <math>1</math>, respectively.
+* Addition: <math>a+c\equiv b+d\pmod {m}</math>.
+* Subtraction: <math>a-c\equiv b-d\pmod {m}</math>.
+* Multiplication: <math>ac\equiv bd\pmod {m}</math>.
+* Division: <math>\frac{a}{e}\equiv \frac{b}{e}\pmod {\frac{m}{\gcd(m,e)}}</math>, where <math>e</math> is a positive integer that divides <math>{a}</math> and <math>b</math>.
+* Exponentiation: <math>a^e\equiv b^e\pmod {m}</math> where <math>e</math> is a positive integer.
-We can use this method to compute <math>M^e</math> modulo <math>n</math>, for any integers <math>M</math> and <math>e</math>, with <math>e > 0</math>.  The beauty of this algorithm is that the process takes, at most, approximately <math>2 \log_2 e</math> steps -- at most <math>\log_2 e</math> steps to compute the values <math>M^k</math> modulo <math>n</math> for <math>k</math> a power of two less than <math>e</math>, and at most <math>\log_2 e</math> steps to multiply the appropriate powers of <math>M</math> according to the binary representation of <math>e</math>.
+== Problem Applications ==
-This method can be further refined using [[Euler's Totient Theorem]].
+https://artofproblemsolving.com/wiki/index.php/2011_AMC_8_Problems/Problem_22
 == Applications of Modular Arithmetic ==
@@ Line 215: / Line 302: @@
 == Resources ==
-* The AoPS [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=10 Introduction to Number Theory] by [[Mathew Crawford]].
+* The AoPS [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:nt Introduction to Number Theory] by [[Mathew Crawford]].
+* The AoPS [http://www.artofproblemsolving.com/School/courseinfo.php?course_id=intro:numbertheory Introduction to Number Theory Course].  Thousands of students have learned more about modular arithmetic and [[problem solving]] from this 12 week class.
 == See also ==
 * [[Intermediate modular arithmetic]]
 * [[Olympiad modular arithmetic]]
+[[Category:Introductory Mathematics Topics]]

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Difference between revisions of "Modular arithmetic/Introduction"

Latest revision as of 07:56, 10 July 2024

Contents

Introductory Video

Understand Modular Arithmetic

Residue

Congruence

Examples

Sample Problem

Solution:

Another Solution:

Another Solution:

Making Computation Easier

Addition

Problem

Solution

Why we only need to use remainders

Solution using modular arithmetic

Addition rule

Proof of the addition rule

Subtraction

Problem

Solution

Subtraction rule

Multiplication

Problem

Solution

Solution using modular arithmetic

Multiplication rule

Exponentiation

Problem #1

Problem #2

Problem #3

Summary of Useful Facts

Problem Applications

Applications of Modular Arithmetic

Resources

See also