the addition or a product of two numbers can result in a odd number in the following 2 circumstances

even+odd= odd

odd+even= odd

odd*odd = odd

consecutive integers are in the format

odd,even,odd

even,odd,even

abc= (a*b)*c = ab must be even hence abc must be even

If a,b,c are in the form odd,even,odd for eg. 1,2,3 then a+b = odd and c will be odd therefore a+b+c will be even

If a,b,c are in the form even,odd,even eg. 2,3,4 then, bc=even and a+bc =even

If If a,b,c are in the form even,odd,even eg.2,3,4 then,a(b+c) = 2(3+4) = 14=even

(a+b)(b+c) follows (odd+even)(even+odd) or (even+odd)(odd+even) in both scenarios the final product will be odd*odd = odd

(E)

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This is my response to the question and may be incorrect. Feel free to rectify any mistakes