Question

Given that 2²⁰ + 1 is completely divisible by a whole number, which of the following is completely divisible by the same number?
1. 5 × 2³⁰
2. 2⁹⁰ + 1
3. 2¹⁵ + 1
4. 2⁶⁰ + 1

Answer 1

Correct Answer - Option 4 : 2⁶⁰ + 1

Given :

2²⁰ + 1 is completely divisible by a whole number 

Formula used :

a³ + b³ = (a + b)(a² + b² - ab)

Calculations :

We will expand the numbers 

2⁹⁰ + 1 = 2⁹⁰ + 1⁹⁰ = (2³⁰ + 1)(2⁶⁰ + 1⁶⁰ - 2³⁰)

2¹⁵ + 1 = 2¹⁵ + 1¹⁵ = (2⁵ + 1) (2¹⁰ + 1¹⁰ - 2⁵)

2⁶⁰ + 1 = 2⁶⁰ + 1⁶⁰ = (2²⁰ + 1)(2⁴⁰ + 1⁴⁰ - 2²⁰)

We can see that in if a number divides (220 + 1) then it will surely divide (260 + 1)

∴ The answer will be the option D