Difference between revisions of "Linear congruence"
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===Solution 1=== | ===Solution 1=== | ||
− | <math>7\equiv15\pmod8</math>, so <math> | + | <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. |
===Solution 2=== | ===Solution 2=== |
Revision as of 05:06, 16 June 2008
A Linear Congruence is a congruence mod p of the form where , , , and are constants and is the variable to be solved for.
Contents
[hide]Solving
Note that not every linear congruence has a solution. For instance, the congruence equation has no solutions. A solution is guaranteed iff is relatively prime to . If and are not relatively prime, let their greatest common divisor be ; then:
- if divides , there will be a solution
- if does not divide , there will be no solution
Example
Problem
Given , find .
Solution 1
, so . Thus, . Note that we can divide by because and are relatively prime.
Solution 2
Multiply both sides of the congruence by to get . Since and , .