More DefinitionsDefinitionSupposeaandbare integers. We sayadividesb, writtena|bifb=acfor somec2Z. In this case we also say thatais adivisorofb, andbis amultipleofa.Example:10 = 2·5I2|10, 5|10I2 is a divisor of 10I10 is a multiple of 2IIs 0 a divisor of 10?No. There is noa2Zwith 0a= 10.We write 06|10IIs-1 a divisor of 10?Notes3
Direct Proof of Conditional StatementsHow to proveP)QProposition:IfP, thenQ.Proof: SupposeP....ThereforeQ.Every step should be completely justified.PQP)QTTTTFFFTTFFTDirect ProofProve the following:If4|x, thenx2is even.Proposition:If 4|x, thenx2is even.Proof: Suppose 4|x.Then by the definition of “divides,” there exists someb2Zsuch that4b=x.Thenx2=4b2= 2bTherefore,x2= 2abfor someab2Z.Therefore,x2is evenby the definition of “even”.Thenthere exists somea2Zsuch that 4a=x. Thenx2= 2a, sox2is even.Direct ProofProve the following:Ifxis odd, thenx2+ 1is even.Proposition:Ifxis odd, thenx2+ 1 is even.Proof: Supposexis odd.Thenx= 2a+ 1 for some integera.Thenx2+1 = (2a+1)2+1 = 4a2+4a+1+1 = 2(2a2+2a+1).
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Negative and non-negative numbers, Prime number, Divisor, Parity, Evenness of zero