Divisibility

Question

It is being given that (232 + 1) is completely divisible by a whole number. Which of the following numbers is completely divisible by this number?

A.(216 + 1).                            B.(216 - 1).                              C.(7 x 223).                            D.(296 + 1)

Answer 1

Option D.

Assuming 2³² is divisible by some number x.

Now, 2⁹⁶ + 1
= (2³²)³ + 1
= (2³²)³ + 1³
= (2³² + 1) ((2³²)² - 2³² + 1)      [(x³ + y3) =  (x+1)(x² - xy +1)]

Look at the first (2³² + 1), it is divisible by x as established. This the Fourth is completely divisible.