It is given that $${\text{(}}{{\text{2}}^{32}} + 1)$$   is exactly divisible by a certain number, which one of the following is also definitely divisible by the same number ?

Solution (By Examveda Team)

$$\eqalign{ & {{\text{a}}^3} + {b^3} = (a + b)({{\text{a}}^2} + {b^2} - ab) \cr & {\text{Take option (A)}} \cr & {\text{ = }}{{\text{2}}^{96}} + 1{\text{ }} \cr & = {({2^{32}})^3} + {(1)^3} \cr & = ({{\text{2}}^{32}} + 1)[{({2^{32}})^2}{\text{ + }}{{\text{1}}^2}{\text{ - }}{{\text{2}}^{32}}] \cr & = ({{\text{2}}^{32}} + 1)[{2^{64}} + 1 - {2^{32}}] \cr & {\text{Clearly, }} \cr & {{\text{2}}^{32}} + 1{\text{ is a factor of }}{{\text{2}}^{96}} + 1 \cr} $$