Week 6 Congruence Relation Modulo n Discrete Math Marie Demlov - PowerPoint PPT Presentation

Congruence Relation Modulo n Residue Classes Modulo n Exercises Week 6 Congruence Relation Modulo n Discrete Math Marie Demlová http://math.feld.cvut.cz/demlova April 2, 2020 M. Demlova: Discrete Math

Congruence Relation Modulo n Residue Classes Modulo n Exercises Congruence Relation Modulo n Given two integers a , b and a natural number n > 1. We say that a is congruent to b modulo n and write a ≡ b ( mod n ) if a − b is divisible by n . Equivalent Characterizations of Modulo n . Let a and b be two integers. Then the following is equivalent: ◮ a ≡ b ( mod n ) , ◮ a = b + k n for some integer k , ◮ a and b have the same remainders when divided by n . M. Demlova: Discrete Math

Congruence Relation Modulo n Residue Classes Modulo n Exercises Congruence Relation Modulo n Proposition. Let a , b , and c be integers. Then ◮ a ≡ a ( mod n ) (modulo n is reflexive); ◮ if a ≡ b ( mod n ) , then also b ≡ a ( mod n ) (modulo n is symmetric); ◮ if a ≡ b ( mod n ) and b ≡ c ( mod n ) , then a ≡ c ( mod n ) (modulo n is transitive). Properties of modulo n . Assume that for integers a , b , c , and d it holds that a ≡ b ( mod n ) and c ≡ d ( mod n ) . Then ( a + c ) ≡ ( b + d ) ( mod n ) a ( a · c ) ≡ ( b · d ) ( mod n ) . M. Demlova: Discrete Math

Congruence Relation Modulo n Residue Classes Modulo n Exercises Congruence Relation Modulo n Corollary. Given two integers a , b such that a ≡ b ( mod n ) . Then ◮ ra ≡ rb ( mod n ) for every integer r ; ◮ a k ≡ b k ( mod n ) for every natural number k . ◮ Moreover, if a i ≡ b i ( mod n ) for every i = 0 , . . . , k , a r 0 , . . . , r k are arbitrary integers, then ( r 0 a 0 + . . . + r k a k ) ≡ ( r 0 b 0 + . . . + r k b k ) ( mod n ) . Proposition. Let r , a , b be integers and n a natural number n > 1 such that ra ≡ rb ( mod n ) . Then � n � a ≡ b mod . gcd( n , r ) M. Demlova: Discrete Math

Congruence Relation Modulo n Residue Classes Modulo n Exercises Congruence Relation Modulo n Solving ( a + x ) ≡ b ( mod n ) . Given integers a , b and a natural number n > 1. Find all integers x for which ( a + x ) ≡ b ( mod n ) . This problem has got always a solution which is any x ∈ Z for which x ≡ ( b − a ) ( mod n ) . Solving ( a · x ) ≡ b ( mod n ) . Given two integers a , b and a natural number n > 1. Find all integers x for which a x ≡ b ( mod n ) . The equation above has a solution iff the number b is a multiple of gcd( a , n ) , and all integers x are solutions of the following Diophantic equation a x + n y = b . M. Demlova: Discrete Math

Congruence Relation Modulo n Residue Classes Modulo n Exercises Congruence Relation Modulo n Proposition. Let n > 1, m > 1 be two relatively prime natural number. And let for some a , b ∈ Z it holds that a ≡ b ( mod n ) and a ≡ b ( mod m ) Then also a ≡ b ( mod nm ) . A stronger version holds: Assume that a ≡ b ( mod n ) and n m a ≡ b ( mod m ) . Let n 1 = gcd( n , m ) and m 1 = gcd( n , m ) . Then a ≡ b ( mod n 1 m 1 ) . Small Fermat Theorem. Let p be a prime and a an integer relatively prime to p . Then a p − 1 ≡ 1 ( mod p ) . M. Demlova: Discrete Math

Congruence Relation Modulo n Residue Classes Modulo n Operations in Z n Exercises Residue Classes Modulo n An equivalence class of the equivalence modulo n containing a number i ∈ Z is the residue class containing i and is denoted by [ i ] n . We have [ i ] n = { j | j = i + kn for some k ∈ Z } . The Set Z n . There are n distinct residue classes modulo n ; indeed, they are the residue classes corresponding to the numbers (remainders) 0 , 1 , . . . , n − 1. The set of all residue classes is denoted by Z n , so Z n = { [ 0 ] n , [ 1 ] n , . . . , [ n − 1 ] n } . M. Demlova: Discrete Math

Congruence Relation Modulo n Residue Classes Modulo n Operations in Z n Exercises Operations in Z n Addition ⊕ and multiplication ⊙ . For [ i ] n , [ j ] n ∈ Z n we have [ i ] n ⊕ [ j ] n = [ i + j ] n , [ i ] n ⊙ [ j ] n = [ i · j ] n . Example. Let n = 6, then there are 6 distinct residue classes, i.e. Z 6 = { [ 0 ] 6 , [ 1 ] 6 , . . . , [ 5 ] 6 } . Moreover, [ 3 ] 6 ⊕ [ 5 ] 6 = [ 2 ] 6 , [ 3 ] 6 ⊙ [ 4 ] 6 = [ 0 ] 6 . M. Demlova: Discrete Math

Congruence Relation Modulo n Residue Classes Modulo n Operations in Z n Exercises Operations in Z n Properties of ⊕ . ◮ ⊕ is associative, i.e. for any three integers i , j , k we have: ([ i ] n ⊕ [ j ] n ) ⊕ [ k ] n = [ i ] n ⊕ ([ j ] n ⊕ [ k ] n ) . ◮ ⊕ is commutative, i.e. for any two integers i , j we have: [ i ] n ⊕ [ j ] n = [ j ] n ⊕ [ i ] n . ◮ The class [ 0 ] n plays the role of “zero”, more precisely, for any integer i we have: [ 0 ] n ⊕ [ i ] n = [ i ] n . ◮ We can ”subtract”, more precisely for any integer [ i ] n there exists class − [ i ] n such that [ i ] n ⊕ ( − [ i ] n ) = [ 0 ] n . M. Demlova: Discrete Math

Congruence Relation Modulo n Residue Classes Modulo n Operations in Z n Exercises Residue Classes Modulo n Properties of the Operation ⊙ . ◮ ⊙ is associative, i.e for any three integers i , j , k we have: ([ i ] n ⊙ [ j ] n ) ⊙ [ k ] n = [ i ] n ⊙ ([ j ] n ⊙ [ k ] n ) . ◮ ⊙ is commutative, i.e. for any two integers i , j we have: [ i ] n ⊙ [ j ] n = [ j ] n ⊙ [ i ] n . ◮ The class [ 1 ] n plays the role of “identity”, More precisely, for any integer i we have: [ 1 ] n ⊙ [ i ] n = [ i ] n . For a residue class [ i ] n there is a residue class [ x ] n such that [ i ] n ⊙ [ x ] n = [ 1 ] n iff the numbers i and n are relatively prime. M. Demlova: Discrete Math

Congruence Relation Modulo n Residue Classes Modulo n Operations in Z n Exercises Residue Classes Modulo n Convention. Later on we will write Z n = { 0 , 1 , . . . , n − 1 } instead of Z n = { [ 0 ] n , [ 1 ] n , . . . , [ n − 1 ] n } and the operations ⊕ , ⊙ will be denoted by an “ordinary signs”, i.e. simply by + and · . Note that we can write that in Z n for every i , j ∈ Z n i + j = k , where k is the remainder when i + j is divided by n ; i · j = l , where l is the remainder when i j is divided by n . M. Demlova: Discrete Math

Congruence Relation Modulo n Residue Classes Modulo n Exercises Exercises Exercies 1. Find all natural numbers x , 0 ≤ x < 555 for which 233 x ≡ 5 ( mod 555 ) . Exercies 2. In Z 414 find all x for which 152 x = 6 . M. Demlova: Discrete Math

Congruence Relation Modulo n Residue Classes Modulo n Exercises Exercises Exercise 3. Find the remainder when you divide 13 742 − 10 · 14 521 + 22 102 . by 7. Exercise 4. Derive and prove criteria for divisibility by 7 and 11. Exercise 5. Write down the multiplication table for ( Z 10 , ⊙ ) . M. Demlova: Discrete Math

Congruence Relation Modulo n Residue Classes Modulo n Exercises Exercises Exercise 6. Find all invertible elements in ( Z 11 , ⊙ ) and their inverses. Exercise 7. Find all invertible elements in ( Z 12 , ⊙ ) and their inverses. M. Demlova: Discrete Math