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Difference between revisions of "Linear congruence"
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Line 13: Line 13:
  
 
<math>5x\equiv 7\equiv 15\pmod{8}</math>, so  
 
<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>5x\equiv 7 \pmod{8}</math>, so
  
 
<math>5\cdot5x\equiv 5\cdot7\pmod{8}</math> and thus
 
<math>5\cdot5x\equiv 5\cdot7\pmod{8}</math> and thus
  
 
<math>x \equiv 3 \pmod 8</math>.
 
<math>x \equiv 3 \pmod 8</math>.
 +
  
  
 
Note that not every linear congruence has a solution.  For instance, the congruence equation
 
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:
+
<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> [[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.
 
* if <math>d</math> does not divide <math>b</math>, there will be no solution.

Revision as of 10:50, 15 August 2006

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.


Example I: How to solve

Say $5x\equiv 7\pmod{8}$. Find $x$.

Solution

$5x\equiv 7\equiv 15\pmod{8}$, so

$x \equiv 3 \pmod 8$. Note that we can divide by 5 because 5 and 8 are relatively prime.

An alternative method is to note that $5x\equiv 7 \pmod{8}$, so

$5\cdot5x\equiv 5\cdot7\pmod{8}$ and thus

$x \equiv 3 \pmod 8$.


Note that not every linear congruence has a solution. For instance, the congruence equation $2x \equiv 3 \pmod 8$ has no solutions. A solution is guaranteed if and only if $a$ is relatively prime to $p$. If $a$ and $p$ are not relatively prime, say with greatest common divisor $d$, then we have two options:

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