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32a-Modular-Arithmetic

# 32a-Modular-Arithmetic - Modular Discrete Mathematics...

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Modular Arithmetic Discrete Mathematics

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Discrete Mathematics – Modular Arithmetic 32-2 Previous Lecture Common divisors The greatest common divisor Euclidean algorithm Least common multiple
Discrete Mathematics – Modular Arithmetic 32-3 Relatively Prime Numbers a and b such that gcd(a,b) = 1 are called relatively prime How many relatively prime numbers are there? Euler’s totient function φ (n) is the number of numbers k such that 0 < k < n and n and k are relatively prime. If p is prime then every k < p is relatively prime with n. Hence, φ (p) = p – 1. Lemma . If a and b are relatively prime then φ (ab) = φ (a) φ (b) Corollary . If is the prime factorization of n, then

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Discrete Mathematics – Modular Arithmetic 32-4 Congruences In some situations we care only about the remainder of an integer when it is divided by some specified positive number. For instance, when we ask what time it will be 50 hours from now, we care only about the remainder of 50 plus the current hour divided by 24. If a and b are integers and m is a positive integer, then a is congruent to b modulo m if m divides a – b. We use the notation a b (mod m) to indicate that a is congruent to b modulo m.
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32a-Modular-Arithmetic - Modular Discrete Mathematics...

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