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  • What is linear congruence with example?

    A congruence of the form ax?b(mod m) where x is an unknown integer is called a linear congruence in one variable. It is important to know that if x0 is a solution for a linear congruence, then all integers xi such that xi?x0(mod m) are solutions of the linear congruence.

  • Different Methods to Solve Linear Congruences

    Example: Solve the linear congruence ax = b (mod m)Solution: ax = b (mod m) _____ (1)Example: Solve the linear congruence 3x = 12 (mod 6)Solution:Example: Solve the Linear Congruence 11x = 1 mod 23.Solution: Find the Greatest Common Divisor of the algorithm.
Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Linear Congruences

Bernd Schr

¨oderBernd Schr

¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruences Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Introduction 1. Linear equations, that is, equations of the form ax=bare

the simplest type of equation we can encounter.2.In this presenta tion,we will consider what happens when

equality is replaced with congruence modulom.3.W ewill also s eeho wthe solution of multiple v erysimple

equations of this type leads to the Chinese Remainder Theorem, which is important because it paves the way for efficiently working with large numbers.4.W ewill conclude w ithsome results about numbers of the form 2 n1, because these numbers are often used as the moduli in computer arithmetics.Bernd Schr ¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruences Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Introduction 1. Linear equations, that is, equations of the form ax=bare

the simplest type of equation we can encounter.2.In this presenta tion,we will consider what happens when

equality is replaced with congruence modulom.3.W ewill also s eeho wthe solution of multiple v erysimple

equations of this type leads to the Chinese Remainder Theorem, which is important because it paves the way for efficiently working with large numbers.4.W ewill conclude w ithsome results about numbers of the form 2 n1, because these numbers are often used as the moduli in computer arithmetics.Bernd Schr ¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruences Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Introduction 1. Linear equations, that is, equations of the form ax=bare

the simplest type of equation we can encounter.2.In this presenta tion,we will consider what happens when

equality is replaced with congruence modulom.3.W ewill also s eeho wthe solution of multiple v erysimple

equations of this type leads to the Chinese Remainder Theorem, which is important because it paves the way for efficiently working with large numbers.4.W ewill conclude w ithsome results about numbers of the form 2 n1, because these numbers are often used as the moduli in computer arithmetics.Bernd Schr ¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruences Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Introduction 1. Linear equations, that is, equations of the form ax=bare

the simplest type of equation we can encounter.2.In this presenta tion,we will consider what happens when

equality is replaced with congruence modulom.3.W ewill also s eeho wthe solution of multiple v erysimple

equations of this type leads to the Chinese Remainder Theorem, which is important because it paves the way for efficiently working with large numbers.4.W ewill conclude w ithsome results about numbers of the form 2 n1, because these numbers are often used as the moduli in computer arithmetics.Bernd Schr ¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruences Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Introduction 1. Linear equations, that is, equations of the form ax=bare

the simplest type of equation we can encounter.2.In this presenta tion,we will consider what happens when

equality is replaced with congruence modulom.3.W ewill also s eeho wthe solution of multiple v erysimple

equations of this type leads to the Chinese Remainder Theorem, which is important because it paves the way for efficiently working with large numbers.4.W ewill conclude w ithsome results about numbers of the form 2 n1, because these numbers are often used as the moduli in computer arithmetics.Bernd Schr ¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruences Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Introduction 1. Linear equations, that is, equations of the form ax=bare

the simplest type of equation we can encounter.2.In this presenta tion,we will consider what happens when

equality is replaced with congruence modulom.3.W ewill also s eeho wthe solution of multiple v erysimple

equations of this type leads to the Chinese Remainder Theorem, which is important because it paves the way for efficiently working with large numbers.4.W ewill conclude w ithsome results about numbers of the form 2 n1, because these numbers are often used as the moduli in computer arithmetics.Bernd Schr ¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruences

Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Theorem.Let m2N, let a;b be integers and let d:= (a;m).

Then the congruence axb(modm)has no solutions iff d-b.In case djb, the congruence has exactly d solutions that are

pairwise incongruent modulo m.Proof.First note thatdjaxfor allx2Z."(."Ifd-b, thend-axbfor allx2Z.Becausemis a

multiple ofd, this implies thatm-axbfor allx2Z.That is,

whend-b, the congruenceaxb(modm)has no solutions.")."We prove the contrapositive, so letdjb.Leta0:=adand let

b 0:=bd .Then(a0;m) =1, so, by earlier theorem, a

01b0;:::;a0mb0is a complete set of residues modulom.Hence, there is ak, so thata0kb00(modm).But then

akb(a0kb0)d0d0(modm).

Bernd Schr

¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruences

Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Theorem.Let m2N, let a;b be integers and let d:= (a;m).

Then the congruence axb(modm)has no solutions iff d-b.In case djb, the congruence has exactly d solutions that are

pairwise incongruent modulo m.Proof.First note thatdjaxfor allx2Z."(."Ifd-b, thend-axbfor allx2Z.Becausemis a

multiple ofd, this implies thatm-axbfor allx2Z.That is,

whend-b, the congruenceaxb(modm)has no solutions.")."We prove the contrapositive, so letdjb.Leta0:=adand let

b 0:=bd .Then(a0;m) =1, so, by earlier theorem, a

01b0;:::;a0mb0is a complete set of residues modulom.Hence, there is ak, so thata0kb00(modm).But then

akb(a0kb0)d0d0(modm).

Bernd Schr

¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruences

Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Theorem.Let m2N, let a;b be integers and let d:= (a;m).

Then the congruence axb(modm)has no solutions iff d-b.In case djb, the congruence has exactly d solutions that are

pairwise incongruent modulo m.Proof.First note thatdjaxfor allx2Z."(."Ifd-b, thend-axbfor allx2Z.Becausemis a

multiple ofd, this implies thatm-axbfor allx2Z.That is,

whend-b, the congruenceaxb(modm)has no solutions.")."We prove the contrapositive, so letdjb.Leta0:=adand let

b 0:=bd .Then(a0;m) =1, so, by earlier theorem, a

01b0;:::;a0mb0is a complete set of residues modulom.Hence, there is ak, so thata0kb00(modm).But then

akb(a0kb0)d0d0(modm).

Bernd Schr

¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruences

Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Theorem.Let m2N, let a;b be integers and let d:= (a;m).

Then the congruence axb(modm)has no solutions iff d-b.In case djb, the congruence has exactly d solutions that are

pairwise incongruent modulo m.Proof.First note thatdjaxfor allx2Z."(."Ifd-b, thend-axbfor allx2Z.Becausemis a

multiple ofd, this implies thatm-axbfor allx2Z.That is,

whend-b, the congruenceaxb(modm)has no solutions.")."We prove the contrapositive, so letdjb.Leta0:=adand let

b 0:=bd .Then(a0;m) =1, so, by earlier theorem, a

01b0;:::;a0mb0is a complete set of residues modulom.Hence, there is ak, so thata0kb00(modm).But then

akb(a0kb0)d0d0(modm).

Bernd Schr

¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruences

Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Theorem.Let m2N, let a;b be integers and let d:= (a;m).

Then the congruence axb(modm)has no solutions iff d-b.In case djb, the congruence has exactly d solutions that are

pairwise incongruent modulo m.Proof.First note thatdjaxfor allx2Z."(."Ifd-b, thend-axbfor allx2Z.Becausemis a

multiple ofd, this implies thatm-axbfor allx2Z.That is,

whend-b, the congruenceaxb(modm)has no solutions.")."We prove the contrapositive, so letdjb.Leta0:=adand let

b 0:=bd .Then(a0;m) =1, so, by earlier theorem, a

01b0;:::;a0mb0is a complete set of residues modulom.Hence, there is ak, so thata0kb00(modm).But then

akb(a0kb0)d0d0(modm).

Bernd Schr

¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruences

Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Theorem.Let m2N, let a;b be integers and let d:= (a;m).

Then the congruence axb(modm)has no solutions iff d-b.In case djb, the congruence has exactly d solutions that are

pairwise incongruent modulo m.Proof.First note thatdjaxfor allx2Z."(."Ifd-b, thend-axbfor allx2Z.Becausemis a

multiple ofd, this implies thatm-axbfor allx2Z.That is,

whend-b, the congruenceaxb(modm)has no solutions.")."We prove the contrapositive, so letdjb.Leta0:=adand let

b 0:=bd .Then(a0;m) =1, so, by earlier theorem, a

01b0;:::;a0mb0is a complete set of residues modulom.Hence, there is ak, so thata0kb00(modm).But then

akb(a0kb0)d0d0(modm).

Bernd Schr

¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruences

Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Theorem.Let m2N, let a;b be integers and let d:= (a;m).

Then the congruence axb(modm)has no solutions iff d-b.In case djb, the congruence has exactly d solutions that are

pairwise incongruent modulo m.Proof.First note thatdjaxfor allx2Z."(."Ifd-b, thend-axbfor allx2Z.Becausemis a

multiple ofd, this implies thatm-axbfor allx2Z.That is,

whend-b, the congruenceaxb(modm)has no solutions.")."We prove the contrapositive, so letdjb.Leta0:=adand let

b 0:=bd .Then(a0;m) =1, so, by earlier theorem, a

01b0;:::;a0mb0is a complete set of residues modulom.Hence, there is ak, so thata0kb00(modm).But then

akb(a0kb0)d0d0(modm).

Bernd Schr

¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruences

Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Theorem.Let m2N, let a;b be integers and let d:= (a;m).

Then the congruence axb(modm)has no solutions iff d-b.In case djb, the congruence has exactly d solutions that are

pairwise incongruent modulo m.Proof.First note thatdjaxfor allx2Z."(."Ifd-b, thend-axbfor allx2Z.Becausemis a

multiple ofd, this implies thatm-axbfor allx2Z.That is,

whend-b, the congruenceaxb(modm)has no solutions.")."We prove the contrapositive, so letdjb.Leta0:=adand let

b 0:=bd .Then(a0;m) =1, so, by earlier theorem, a

01b0;:::;a0mb0is a complete set of residues modulom.Hence, there is ak, so thata0kb00(modm).But then

akb(a0kb0)d0d0(modm).

Bernd Schr

¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruences

Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Theorem.Let m2N, let a;b be integers and let d:= (a;m).

Then the congruence axb(modm)has no solutions iff d-b.In case djb, the congruence has exactly d solutions that are

pairwise incongruent modulo m.Proof.First note thatdjaxfor allx2Z."(."Ifd-b, thend-axbfor allx2Z.Becausemis a

multiple ofd, this implies thatm-axbfor allx2Z.That is,

whend-b, the congruenceaxb(modm)has no solutions.")."We prove the contrapositive, so letdjb.Leta0:=adand let

b 0:=bd .Then(a0;m) =1, so, by earlier theorem, a

01b0;:::;a0mb0is a complete set of residues modulom.Hence, there is ak, so thata0kb00(modm).But then

akb(a0kb0)d0d0(modm).

Bernd Schr

¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruences

Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Theorem.Let m2N, let a;b be integers and let d:= (a;m).

Then the congruence axb(modm)has no solutions iff d-b.In case djb, the congruence has exactly d solutions that are

pairwise incongruent modulo m.Proof.First note thatdjaxfor allx2Z."(."Ifd-b, thend-axbfor allx2Z.Becausemis a

multiple ofd, this implies thatm-axbfor allx2Z.That is,

whend-b, the congruenceaxb(modm)has no solutions.")."We prove the contrapositive, so letdjb.Leta0:=adand let

b 0:=bd .Then(a0;m) =1, so, by earlier theorem, a

01b0;:::;a0mb0is a complete set of residues modulom.Hence, there is ak, so thata0kb00(modm).But then

akb(a0kb0)d0d0(modm).

Bernd Schr

¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruences

Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Theorem.Let m2N, let a;b be integers and let d:= (a;m).

Then the congruence axb(modm)has no solutions iff d-b.In case djb, the congruence has exactly d solutions that are

pairwise incongruent modulo m.Proof.First note thatdjaxfor allx2Z."(."Ifd-b, thend-axbfor allx2Z.Becausemis a

multiple ofd, this implies thatm-axbfor allx2Z.That is,

whend-b, the congruenceaxb(modm)has no solutions.")."We prove the contrapositive, so letdjb.Leta0:=adand let

b 0:=bd .Then(a0;m) =1, so, by earlier theorem, a

01b0;:::;a0mb0is a complete set of residues modulom.Hence, there is ak, so thata0kb00(modm).But then

akb(a0kb0)d0d0(modm).

Bernd Schr

¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruences

Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Theorem.Let m2N, let a;b be integers and let d:= (a;m).

Then the congruence axb(modm)has no solutions iff d-b.In case djb, the congruence has exactly d solutions that are

pairwise incongruent modulo m.Proof.First note thatdjaxfor allx2Z."(."Ifd-b, thend-axbfor allx2Z.Becausemis a

multiple ofd, this implies thatm-axbfor allx2Z.That is,

whend-b, the congruenceaxb(modm)has no solutions.")."We prove the contrapositive, so letdjb.Leta0:=adand let

b 0:=bd .Then(a0;m) =1, so, by earlier theorem, a

01b0;:::;a0mb0is a complete set of residues modulom.Hence, there is ak, so thata0kb00(modm).But then

akb(a0kb0)d0d0(modm).

Bernd Schr

¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruences

Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Theorem.Let m2N, let a;b be integers and let d:= (a;m).

Then the congruence axb(modm)has no solutions iff d-b.In case djb, the congruence has exactly d solutions that are

pairwise incongruent modulo m.Proof.First note thatdjaxfor allx2Z."(."Ifd-b, thend-axbfor allx2Z.Becausemis a

multiple ofd, this implies thatm-axbfor allx2Z.That is,

whend-b, the congruenceaxb(modm)has no solutions.")."We prove the contrapositive, so letdjb.Leta0:=adand let

b 0:=bd .Then(a0;m) =1, so, by earlier theorem, a

01b0;:::;a0mb0is a complete set of residues modulom.Hence, there is ak, so thata0kb00(modm).But then

akb(a0kb0)d0d0(modm).

Bernd Schr

¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruences

Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Theorem.Let m2N, let a;b be integers and let d:= (a;m).

Then the congruence axb(modm)has no solutions iff d-b.In case djb, the congruence has exactly d solutions that are

pairwise incongruent modulo m.Proof.First note thatdjaxfor allx2Z."(."Ifd-b, thend-axbfor allx2Z.Becausemis a

multiple ofd, this implies thatm-axbfor allx2Z.That is,

whend-b, the congruenceaxb(modm)has no solutions.")."We prove the contrapositive, so letdjb.Leta0:=adand let

b 0:=bd .Then(a0;m) =1, so, by earlier theorem, a

01b0;:::;a0mb0is a complete set of residues modulom.Hence, there is ak, so thata0kb00(modm).But then

akb(a0kb0)d0d0(modm).

Bernd Schr

¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruences

Solving Linear CongruencesChinese Remai nderTheor emNumbers 2n1Theorem.Let m2N, let a;b be integers and let d:= (a;m).

Then the congruence axb(modm)has no solutions iff d-b.In case djb, the congruence has exactly d solutions that are

pairwise incongruent modulo m.Proof.First note thatdjaxfor allx2Z."(."Ifd-b, thend-axbfor allx2Z.Becausemis a

multiple ofd, this implies thatm-axbfor allx2Z.That is,

whend-b, the congruenceaxb(modm)has no solutions.")."We prove the contrapositive, so letdjb.Leta0:=adand let

b 0:=bd .Then(a0;m) =1, so, by earlier theorem, a

01b0;:::;a0mb0is a complete set of residues modulom.Hence, there is ak, so thata0kb00(modm).But then

akb(a0kb0)d0d0(modm).

Bernd Schr

¨oderLouisiana T echUni versity,College of Engineering and Science Linear Congruencesquotesdbs_dbs12.pdfusesText_18
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