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25 Cards in this Set
- Front
- Back
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A set is...
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... an unordered collection of distinct objects
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Let P, Q be mathematical statements. P implies Q is logically equivalent to...
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...Not Q implies Not P
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If a|b and b|c then...
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...a|c
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If a|b and b|a then...
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...a= +/- b
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Let c|A1,...,An. Then for integers B1,...,Bn...
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...c| (A1*B1+...+An*Bn)
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Let a,b be two integers. If d|a,b then we call d...
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...a common divisor
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Let a1,...,ak be integers. Then gcd(a1,...,ak)=...
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...gcd(gcd(a1,a2),a3,...,ak)
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If gcd(a,b)=1, we call a and b...
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...relatively prime
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If a1,...,ak are integers and gcd(a1,...,ak)=1, we call...
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....this list relatively prime
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Let a1,...,ak be integers. If gcd(ai,aj)=1 for all i,j then we call this list...
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...mutually coprime
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Euclidean Division refers to...
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...division with remainder
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(Theorem) Let a,b be integers with b>0. Then there exist...
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...UNIQUE integers q,r such that a=qb+r and 0<r<b
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Let S be a set. What does the well-ordering principle say about S?
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That S contains a smallest element
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(Lemma to the remainder theorem) Let a,b be integers with b>0 and let a=qb+r be the result of a Euclidean Division.Then...
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...gcd(a,b)=gcd(b,r)
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Let a,b be integers. Then ax+by= gcd(a,b)...
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...has a pair of integer solutions (x,y)
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Let a,b be nonzero integers and let c be some other integer. Then ax+by=c has a pair of integer solutions IFF...
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...gcd(a,b)|c. If (x1,y1) is an integer solution, then all solutions are of the form x= x1+(b/d)n, y=y1-(a/d)n, where d is the gcd
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Let a,b be relatively prime and let c be an integer. If a|bc then...
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...a|c
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(Euclid's Lemma) Let p be a prime. If p|(a1*a2*...*an) then...
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...p|aK for some K
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If gcd(a,b)=1 and if a|c and b|c, then...
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...ab|c
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Let a,b be nonzero integers. If c is a common divisor of a and b (c|a and c|b) then...
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...c | gcd(a,b)
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Let a,b be two integers and let m be a positive integer. Then gcd(ma, mb)=...
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...m*gcd(a,b)
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If d|a,b then gcd(a/d, b/d)=...
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...gcd(a,b)/d
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If a,b are integers, the smallest positive integer which is a multiple of both a and b is...
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...called the least common multiple
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gcd(a,b)*lcm(a,b)=...
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...a*b
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If c is a common multiple of a,b, then c...
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...is a multiple of the lcm(a,b)
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