Difference between revisions of "Linear congruence"

Revision as of 00:08, 19 December 2009

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

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 13: / Line 13: @@
 ===Solution 1===
-<math>7\equiv15\pmod8</math>, so <math>5x\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.
+<math>7\equiv15\pmod8</math>, so <math>5x\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. <math>/square</math>
 ===Solution 2===

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