Definition. Let a, b Z . We say that a divides b , denoted a | b , - PDF document

Definition. Let a, b ∈ Z . We say that a divides b , denoted a | b , if there is k ∈ Z such that b = k · a . Then we say that a is a factor of b and that b is a multiple of a .

Fact. For every n ∈ Z the following are true: 1 | n , n | n , and n | 0.

Theorem. Let a, b, c ∈ Z . (i) If a | b and b | c , then a | c . (ii) a | b if and only if | a | | | b | . (iii) If a | b and b � = 0, then | a | ≤ | b | .

Theorem. The relation a | b is a partial ordering on N and on N 0 .

Theorem. (division theorem) Let a, d ∈ Z , d � = 0. Then there exist q ∈ Z and r ∈ N 0 such that a = qd + r and 0 ≤ r < | d | . The numbers q and r are unique. Definition. The number r is called the remainder and we denote it r = a mod d . The number q is called the quotient .

Fact. Let a ∈ Z and d ∈ N , let q be the quotient of a by d . � a � Then q = . d

Fact. Let a, b ∈ Z , a � = 0. Then a | b if and only if b mod | a | = 0, that is, the remainder when dividing b by | a | is 0.

Definition. Let a, b ∈ Z . A number d ∈ N is called a common divisor of numbers a, b if d | a and d | b . A number d ∈ N is called a common multiple of numbers a, b if a | d and b | d .

Definition. Let a, b ∈ Z . We define their greatest common divisor , denoted gcd( a, b ), as the largest element of the set of common divisors, if at least one of a, b is not zero. Otherwise we define gcd(0 , 0) = 0. We say that numbers a, b ∈ Z are coprime if gcd( a, b ) = 1. We define their least common multiple , denoted lcm( a, b ), as the smallest element of the set of their common multiples, if a, b are both not zero. Otherwise we set lcm( a, 0) = lcm(0 , b ) = 0.

Fact. Let a, b ∈ Z . Then gcd( a, b ) = gcd( | a | , | b | ) and lcm( a, b ) = lcm( | a | , | b | ).

Theorem. Let a, b ∈ Z . Then lcm( a, b ) · gcd( a, b ) = | a | · | b | .

Fact. Let a ∈ N . Then gcd( a, 0) = a , lcm( a, 0) = 0 and gcd( a, a ) = lcm( a, a ) = a .

Lemma. Let a > b ∈ N , let q, r ∈ N 0 satisfy a = qb + r and 0 ≤ r < b . Then the following are true: (i) d ∈ N is a common divisor of a, b if and only if it is a common divisor of b, r . (ii) gcd( a, b ) = gcd( b, r ).

Euklid’s algorithm for finding gcd( a, b ) for a > b ∈ N . Version 1. Initiation: r 0 := a , r 1 := b , k := 0. Step: k := k + 1, r k − 1 = q k · r k + r k +1 Repeat until r k +1 = 0. Then gcd( a, b ) = r k . Version 2. procedure gcd ( a, b : integer) repeat r := a mod b ; a := b ; b := r ; until b = 0 ; output: a ;

Theorem. (Bezout theorem/identity) Let a, b ∈ Z . Then there are A, B ∈ Z such that gcd( a, b ) = Aa + Bb .

Extended Euclid algorithm for finding gcd( a, b ) = Aa + Bb for a > b ∈ N . Version 1. Initiation: r 0 := a , r 1 := b , k := 0, A 0 := 1, A 1 := 0, B 0 := 0, B 1 := 1. � r k − 1 � Step: k := k + 1, q k := , r k r k +1 := r k − 1 − q k r k , A k +1 := A k − 1 − q k A k , B k +1 := B k − 1 − q k B k . Repeat until r k +1 = 0. Then gcd( a, b ) = r k = A k a + B k b . Version 2. procedure gcd-Bezout ( a, b : integer) A 0 := 1 ; A 1 := 0 ; B 0 := 0 ; B 1 := 1 ; repeat � a � q := ; b r := a − qb ; r a := A 0 − qA 1 ; r b := B 0 − qB 1 ; a := b ; b := r ; A 0 := A 1 ; A 1 := r a ; B 0 := B 1 ; B 1 := r b ; until b = 0 ; output: gcd( a, b ) = a = A 0 a + B 0 b ;

Definition. By a linear diophantine equation of two variables we mean any equation of the form ax + by = c with unknowns x, y , where a, b, c ∈ Z and only integer solutions are allowed.

Theorem. A linear diophantine equation ax + by = c has at least one solution if and only if c is a multiple of gcd( a, b ).

Theorem. Consider a linear diophantine equation ax + by = c . Let ( x p , y p ) ∈ Z 2 be some particular solution of this equation. A vector ( x, y ) ∈ Z 2 is a solution of this equation if and only if there exists some ( x h , y h ) ∈ Z 2 such that ( x, y ) = ( x p , y p ) + ( x h , y h ) and ( x h , y h ) solves the associated homogeneous equation.

Theorem. Consider an equation ax + by = 0 for a, b ∈ Z . Then the set of all its integer solutions is b a �� � � − k gcd( a, b ) , k ; k ∈ Z . gcd( a, b )

Algorithm for finding all integer solutions of the equation ax + by = c . 0. Find (for instance using the extended Euclid’s algorithm) A, B ∈ Z such that gcd( a, b ) = Aa + Bb . 1. If c is not a multiple of gcd( a, b ), then there is no solution. 2. The case gcd( a, b ) divides c : c a) Multiply the equality aA + bB = gcd( a, b ) by c ′ = gcd( a, b ), keeping coefficients a, b intact, you will obtain a ( Ac ′ ) + b ( Bc ′ ) = c and thus one particular solution x p = Ac ′ , y p = Bc ′ . b) Cancel gcd( a, b ) in the associated homogeneous equation ax + by = 0, obtaining a ′ x + b ′ y = 0, that is, a ′ x = − b ′ y . This gives x h = − b ′ k , y h = a ′ k for k ∈ Z . c) By adding the particular and homogeneous solutions you obtain the general integer solution of the given equation c b x = Ac ′ − b ′ k = A gcd( a, b ) − gcd( a, b ) k, c a y = Bc ′ + a ′ k = B gcd( a, b ) + gcd( a, b ) k ; k ∈ Z .

Algorithm 2 for finding all integer solutions of the equation ax + by = c . 1. Guess gcd( a, b ). Try to cancel this number in the given equation. If it is not possible, that is, if c is not a multiple of gcd( a, b ), then there is no solution. 2. Case gcd( a, b ) divides c : Divide the given equation by gcd( a, b ). You will obtain a new dio- phantine equation a ′ x + b ′ y = c ′ , where a ′ , b ′ are coprime. a) Find (for instance using the extended Euclid’s algorithm) A, B ∈ Z such that gcd( a ′ , b ′ ) = 1 = Aa ′ + Bb ′ , so a ′ A + b ′ B = 1. Multiply this equality by c ′ keeping coefficients intact, you will obtain a ′ ( Ac ′ ) + b ′ ( Bc ′ ) = c ′ and thus one particular solution x p = Ac ′ , y p = Bc ′ , that is, a vector ( Ac ′ , Bc ′ ). b) Solve the associated homogeneous equation a ′ x + b ′ y = 0, that is, a ′ x = − b ′ y , which gives x h = − b ′ k , y h = a ′ k , in other words the pair ( − b ′ k, a ′ k ) for k ∈ Z . c) By adding the particular and homogeneous solutions you obtain the set of all integer solutions { ( Ac ′ − kb ′ , Bc ′ + ka ′ ); k ∈ Z } .

Definition. Let a ∈ N , a � = 1. We say that it is a prime if the only natural numbers that divide it are 1 and a . We say that a is a composite number if it is not a prime.

2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47, 53 , 59 , 61 , 67 , 71 , 73 , 79 , 83 , 89 , 97.

Lemma. Let a 1 , . . . , a m ∈ N and p be a prime. If p | ( a 1 a 2 · · · a m ), then there is i such that p | a i .

Theorem. (Fundamental theorem of arithmetics, prime decompo- sition) Let n ∈ N . Then there exist primes p 1 , p 2 , . . . , p m and exponents k 1 , k 2 , . . . , k m ∈ N 0 so that m n = p k 1 1 · p k 2 p k i 2 · · · p k m m = � i . i =1 If we also demand that p 1 < p 2 < . . . < p m and k i > 0, then this decomposition is unique.