Divisibility, modular arithmetic, and prime numbers from an intro discrete mathematics course.
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- What does it mean for integer a to divide integer b?
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There exists an integer k such that b = ak.- Notation: what does 'a | b' mean?
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a divides b.- Is divisibility reflexive? Does a | a for all nonzero a?
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Yes, a = a * 1, so a divides itself.- Is divisibility transitive? If a | b and b | c, does a | c?
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Yes, b = ak and c = bm for some integers k, m, so c = a(km).- If a | b and a | c, what can you say about a | (b + c)?
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a divides (b + c), since b = ak and c = am implies b + c = a(k + m).- Define the greatest common divisor (GCD) of two nonzero integers a and b.
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The largest positive integer that divides both a and b.- What does gcd(a, b) = 1 mean?
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a and b are coprime (mutually prime).- State Bezout's identity for integers a and b.
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There exist integers x and y such that ax + by = gcd(a, b).- What is the Euclidean algorithm for computing gcd(a, b)?
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Repeatedly divide: gcd(a, b) = gcd(b, a mod b), until the remainder is 0.- Define a prime number.
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An integer p > 1 with no positive divisors other than 1 and p itself.- Define a composite number.
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An integer n > 1 that is not prime, i.e., has a divisor other than 1 and itself.- State the Fundamental Theorem of Arithmetic.
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Every integer n > 1 can be uniquely factored as a product of primes, up to order.- What does 'a ≡ b (mod n)' mean?
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n divides (a - b), or equivalently, a and b have the same remainder when divided by n.- Is congruence modulo n reflexive? Does a ≡ a (mod n)?
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Yes, n | 0 = (a - a).- Is congruence modulo n symmetric? If a ≡ b (mod n), does b ≡ a (mod n)?
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Yes, if n | (a - b), then n | (b - a).
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