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In [[modular arithmetic]], the question of when a linear [[modular arithmetic|congruence]] can be solved is answered by the '''linear  congruence theorem'''. If ''a'' and ''b'' are any [[integer]]s  and ''n'' is a positive integer, then the congruence
:<math>ax \equiv b \pmod {n}</math>
has a solution for ''x'' if and only if ''b'' is divisible by the  [[greatest common divisor]] ''d'' of ''a'' and ''n'' (denoted by gcd(''a'',''n'')). When this is the case, and ''x''<sub>0</sub> is one solution of (1), then the set of all solutions is given by
 
:<math>\{x_0+k\frac{n}{d}\mid k\in\Bbb{Z}\}.</math>
 
In particular, there will be exactly ''d'' = gcd(''a'',''n'') solutions in the set of residues {0,1,2,...,''n''&nbsp;&minus;&nbsp;1}. The result is a simple consequence of [[Bézout's identity]].
 
==Example==
For example, examining the equation a''x'' ≡ 2 ('''mod''' 6)  with different values of ''a'' yields
:<math>3x \equiv 2 \pmod 6\ </math>
Here d = gcd(3,6) = 3 but since 3 does not divide 2, there is no  solution.  
:<math>5x \equiv 2 \pmod 6\ </math>
Here ''d'' = gcd(5,6) = 1, which divides any ''b'', and so there is just one solution in {0,1,2,3,4,5}: ''x''&nbsp;=&nbsp;4.
:<math>4x \equiv 2 \pmod {6}\ </math>
Here ''d'' = gcd(4,6) = 2, which does divide 2, and so there are exactly two solutions in {0,1,2,3,4,5}: ''x''&nbsp;=&nbsp;2 and ''x''&nbsp;=&nbsp;5.
 
==Solving a linear congruence==
In general solving equations of the form:
:<math>ax \equiv b \pmod {n}</math>
If the greatest common divisor ''d'' = gcd(''a'', ''n'') divides ''b'', then we can find a solution ''x'' to the congruence as follows:  the [[extended Euclidean algorithm]] yields integers ''r'' and ''s'' such ''ra'' + ''sn'' = ''d''. Then ''x'' = ''rb''/''d'' is a solution. The other solutions are the numbers congruent to ''x'' modulo ''n''/''d''.
 
For example, the congruence
:<math>12x \equiv 20 \pmod {28}\ </math>
has 4 solutions since gcd(12, 28) = 4 divides 20. The extended  Euclidean algorithm gives (&minus;2)&middot;12 + 1&middot;28 = 4, i.e. ''r'' = &minus;2 and  ''s'' = 1. Therefore, one solution is ''x'' = &minus;2&middot;20/4 = &minus;10, and &minus;10 = 4 modulo 7. All other solutions will also be congruent to 4 modulo 7. Since the original equation uses modulo 28, the entire solution set in the range from 0 to 27 is {4, 11, 18, 25}.
 
==System of linear congruences==
By repeatedly using the linear congruence theorem, one can also solve systems of linear congruences, as in the following example: find all numbers ''x'' such that
:<math>2x \equiv 2 \pmod {6}</math>
:<math>3x \equiv 2 \pmod {7}</math>
:<math>2x \equiv 4 \pmod {8}</math>
By solving the first congruence using the method explained above, we find <math>x \equiv 1 \pmod{3}</math>, which can also be written as <math>x = 3k + 1</math>. Substituting this into the second congruence and simplifying, we get
:<math>9k \equiv - 1 \pmod{7}</math>.
Solving this congruence yields <math>k \equiv 3 \pmod{7}</math>, or <math>k = 7l + 3</math>. It then follows that <math>x = 3 (7l + 3) + 1 = 21l + 10</math>. Substituting this into the third congruence and simplifying, we get
:<math>42l \equiv - 16 \pmod{8}</math>
which has the solution <math>l \equiv 0 \pmod{4}</math>, or <math>l = 4m</math>. This yields <math>x = 21(4m) + 10 = 84m + 10</math>, or
:<math>\equiv 10 \pmod{84}</math>
which describes all solutions to the system.
 
==See also==
*[[Chinese remainder theorem]]
 
{{DEFAULTSORT:Linear Congruence Theorem}}
[[Category:Modular arithmetic]]
[[Category:Theorems in number theory]]

Latest revision as of 02:09, 11 January 2015



What if you could reach thousands of prospects, build fruitful, pro-active relationships with them, and also top-of-mind with them on an authentic basis.and not spend a fortune in the process?

Upon leaving Salt Lake City, UT, Kody's home - Kody saw his brother Kris down the fishing line moving some company vehicles around. Being a hurry, Kody ignored a PROMPTING to work over and say goodbye to Kris. Running late and from a rush, Kody honked the moving truck's horn and waved goodbye to his brother.

No need to panic and begin to change dietary habits, but professional to add the as well as vitamins food following EABC foods containing antioxidants substances.

Since, this sophisticated widget supports GPRS and EDGE, accessing internet becomes easy as well as rapidly. With the help worth mentioning technologies building internet connection is considerably more hassle easy. Users can download wallpapers, music tracks, games a number of more other files at fastest possible speed. Mails can be send with heavy attachments at superb speed. Moreover, you can access Web from any corner of the world. This method for you to be linked to your work or for ones near or dear ones anytime as well as at any make.

The system relies on advanced technology developed a new group of 31 experienced traders, continuously developed until it produced 100% accurate results perpetually. As the Forex market fluctuates many times a day, This Forex Signals system sends out entry signals (buy or sell) to the members, either by email or by SMS. These trading signals occurs genuine time, as the currency variation.

Next, search the professional and business media for information. Talk to reporters who write because of the organization. You'll quickly identify the trends, and you'll develop inquiries to ask .

The more backlinks should get, far better. So not solely these days reverse backlinks in invest other persons are not counted a excessive value backlink any extra, it will also be pretty time consuming to undertake it that strategy. The internet has exploded, and so has the volume websites. And all of them preventing for top rankings. This provides the explanation why the whole lot should be blown up much more today. Which is just the way it is considered to be. Fortuitously, not solely competitors has multiplied. But additionally technology has grown. The net advertising strategies turned more intelligent.

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