Modular Arithmetic
... • If a, b, n ∈ Z and n 6= 0, a is congruent to b mod n if n | a − b. Notation: a = b (mod n). • Congruence mod n is an equivalence relation on Z. • Zn is the set of equivalence classes of Z under the relation of congruence mod n. • Every integer is congruent mod n to a unique integer in {0, 1, . . . ...
... • If a, b, n ∈ Z and n 6= 0, a is congruent to b mod n if n | a − b. Notation: a = b (mod n). • Congruence mod n is an equivalence relation on Z. • Zn is the set of equivalence classes of Z under the relation of congruence mod n. • Every integer is congruent mod n to a unique integer in {0, 1, . . . ...
Euclid`s algorithm and multiplicative inverse
... So the solutions are those values of x that are congruent to 7 modulo 18. In particular, (13−1 mod 18) = 7. (Verify!) REMARK. Let us review the logic of this argument. We started from the pair of congruences (3) and from these we deduced congruence (6). Does this mean that x is a multiplicative inve ...
... So the solutions are those values of x that are congruent to 7 modulo 18. In particular, (13−1 mod 18) = 7. (Verify!) REMARK. Let us review the logic of this argument. We started from the pair of congruences (3) and from these we deduced congruence (6). Does this mean that x is a multiplicative inve ...
A lgebraic Solution of the C oincidence Problem in Two
... tions of given index m can be collected into equiva lence classes of rotations related by the action of O [24]. This double coset analysis will be described in [23]. For example, truly different CSL's of Z 3 with the same index occur for the first time at U = 13. Also, describing the fine structure ...
... tions of given index m can be collected into equiva lence classes of rotations related by the action of O [24]. This double coset analysis will be described in [23]. For example, truly different CSL's of Z 3 with the same index occur for the first time at U = 13. Also, describing the fine structure ...
Section2.2notesall
... The common prime factors of two numbers can be used to find the greatest common divisor of two numbers, which we define next. Definition: The greatest common divisor of two natural numbers a and b , denoted as gcd( a, b) , is the largest natural number that divides a and b with no remainder. Elemen ...
... The common prime factors of two numbers can be used to find the greatest common divisor of two numbers, which we define next. Definition: The greatest common divisor of two natural numbers a and b , denoted as gcd( a, b) , is the largest natural number that divides a and b with no remainder. Elemen ...
Algorithmic Number Theory
... Fact 2.1 The following are easy to show. 1. 1|a for all a ∈ Z, 2. a|a for all a 6= 0, 3. a|b implies a|bc, for all c ∈ Z, 4. a|b and b|c implies a|c, 5. a|b and a|c implies a|b ± c, 6. Every prime is a positive integer. 2 is the smallest prime. Theorem 2.2 The set of primes is infinite. Proof outlin ...
... Fact 2.1 The following are easy to show. 1. 1|a for all a ∈ Z, 2. a|a for all a 6= 0, 3. a|b implies a|bc, for all c ∈ Z, 4. a|b and b|c implies a|c, 5. a|b and a|c implies a|b ± c, 6. Every prime is a positive integer. 2 is the smallest prime. Theorem 2.2 The set of primes is infinite. Proof outlin ...
