(Solved): 1) Prove that the product of any two consecutive integers is even. 2) Suppose ...

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1) Prove that the product of any two consecutive integers is even. 2) Suppose $$\mathrm{b}$$ is an integer. If $$\mathrm{b}\mathrm{mod}12=5$$, what is $$8?\mathrm{b}\mathrm{mod}12$$ ? In other words, if division of $$\mathrm{b}$$ by 12 gives a remainder of 5 , what is the remainder when $$8?\mathrm{b}$$ is divided by 12 ? 3) Seven pounds of raw material are needed to manufacture each unit of a certain product. Express the number of units that can be produced from n pounds of raw material using either the floor or the ceiling notation. Which notation is more appropriate?