Difference between revisions of "Linear congruence"

Revision as of 19:53, 25 April 2008

A Linear Congruence is a congruence mod p of the form $ax+b\equiv c\pmod{p}$ where $a$ , $b$ , $c$ , and $p$ are constants and $x$ is the variable to be solved for.

Solving

Note that not every linear congruence has a solution. For instance, the congruence equation $2x\equiv3\pmod8$ has no solutions. A solution is guaranteed iff $a$ is relatively prime to $p$ . If $a$ and $p$ are not relatively prime, let their greatest common divisor be $d$ ; then:

if $d$ divides $b$ , there will be a solution $\pmod{\frac{p}{d}}$
if $d$ does not divide $b$ , there will be no solution

Example

Problem

Given $5x\equiv7\pmod8$ , find $x$ .

Solution 1

$7\equiv15\pmod8$ , so $3x\equiv15\pmod8$ . Thus, $x\equiv3\pmod 8$ . Note that we can divide by $5$ because $5$ and $8$ are relatively prime.

Solution 2

Multiply both sides of the congruence by $5$ to get $25x\equiv35\pmod8$ . Since $25\equiv1\pmod8$ and $35\equiv3\pmod8$ , $x\equiv3\pmod8$ .

@@ Line 1: / Line 1: @@
-A '''Linear Congruence''' is a [[congruence]] mod p of the form
+A '''Linear Congruence''' is a [[congruence]] [[modular arithmetic|mod]] p of the form
+<cmath>ax+b\equiv c\pmod{p}</cmath>
+where <math>a</math>, <math>b</math>, <math>c</math>, and <math>p</math> are [[constant]]s and <math>x</math> is the [[variable]] to be solved for.
-<math>ax+b\equiv c\pmod{p}</math>
+==Solving==
+Note that not every linear congruence has a solution. For instance, the congruence equation <math>2x\equiv3\pmod8</math> has no solutions. A solution is guaranteed [[iff]] <math>a</math> is [[relatively prime]] to <math>p</math>. If <math>a</math> and <math>p</math> are not relatively prime, let their [[greatest common divisor]] be <math>d</math>; then:
+* if <math>d</math> [[divide]]s <math>b</math>, there will be a solution <math>\pmod{\frac{p}{d}}</math>
+* if <math>d</math> does not divide <math>b</math>, there will be no solution
-where <math>a, b, c</math>, and <math>p</math> are [[constant]]s and <math>x</math> is the [[variable]] to be solved for.
+==Example==
+===Problem===
+Given <math>5x\equiv7\pmod8</math>, find <math>x</math>.
+===Solution 1===
+<math>7\equiv15\pmod8</math>, so <math>3x\equiv15\pmod8</math>. Thus, <math>x\equiv3\pmod 8</math>. Note that we can divide by <math>5</math> because <math>5</math> and <math>8</math> are relatively prime.
-==Example I: How to solve==
+===Solution 2===
+Multiply both sides of the congruence by <math>5</math> to get <math>25x\equiv35\pmod8</math>. Since <math>25\equiv1\pmod8</math> and <math>35\equiv3\pmod8</math>, <math>x\equiv3\pmod8</math>.
-Say <math>5x\equiv 7\pmod{8}</math>.  Find <math>x</math>.
+[[Category:Algebra]]
-===Solution===
-<math>5x\equiv 7\equiv 15\pmod{8}</math>, so
-<math>x \equiv 3 \pmod 8</math>.  Note that we can divide by 5 because 5 and 8 are [[relatively prime]].
-An alternative method is to note that
-<math>5\cdot 5 \equiv 25 \equiv 1 \pmod{8}</math>, so
-<math>5\cdot5x\equiv 5\cdot7\pmod{8}</math> and thus
-<math>x \equiv 3 \pmod 8</math>.
-Note that not every linear congruence has a solution.  For instance, the congruence equation
-<math>2x \equiv 3 \pmod 8</math> has no solutions.  A solution is guaranteed if and only if <math>a</math> is relatively prime to <math>p</math>.  If <math>a</math> and <math>p</math> are not relatively prime, say with [[greatest common divisor]] <math>d</math>, then we have two options:
-* if <math>d</math> [[divide]]s <math>b</math>, there will be a solution <math>\pmod{ \frac{p}{d}}</math>
-* if <math>d</math> does not divide <math>b</math>, there will be no solution.

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