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Let a,b∈ℤ. We say that a divides b, if there exists c∈ℤ s.t. b=ac. We write a∣b if a divides b, and a∤b if a does not divide b.
Ex. 3|6 since there exists c in Z st 6=3c. Here c=2. Hence 3 is a divisor of 6.
a|0 (c=0) and 0|a iff a=0 |
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a|b is not the same as a/b or b/a. a|b is a statement about the relationship between two integers: it says that a divides into b evenly with no remainder.
a|b = statement ABOUT numbers while a/b ARE numbers |
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Prop 1.1 (Transitive)
Let a,b,c∈ℤ. If a|b and b|c,
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then a|c.
Proof: since a|b and b|c, there exists e,f∈ℤ st b=ae and c=bf. Then c=bf=(ae)f=a(ef) and a|c. |
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Prop 1.2
Let a,b,c,m,n be in Z. If c|a and c|b, |
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then c|ma+nb.
This says that an integer dividing each of two integers also divides any integral linear combination of those two integers.
Proof: Since c|a and c|b, there exists e,f in Z st a=ce and b=cf. Then ma+nb=mce+ncf=c(me+nf)
and c|ma+nb. |
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| Expression: integral linear combination of a and b |
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| Def: greatest integer function of x |
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| Let x be in R. The greatest integer function of x, denoted [x] (no top part), is the greatest integer less than or equal to x. |
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| Let x be in R. Then x-1<[x]\ |
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Let a,b be in Z with b>0. Then there exist unique q,r in Z st
a=bq+r, 0≤r less than b (q=quotient, r=remainder) |
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