Question

If N and x are positive integers such that N^N = 2¹⁶⁰ and N² + 2^N is an integral multiple of 2^x, then the largest possible x is

Answer 1

Calculation:

N^N = 2¹⁶⁰ = 2^{5 × 32}

⇒ (2⁵)³²

⇒ (32)³²; which is in form of N^N

⇒ N²+ 2^N = 32² + 2³²

⇒ (2⁵)² + 2³²

⇒ 2¹⁰ + 2³²

⇒ 2¹⁰(1 + 2²²)

⇒ N²+ 2^N = 2¹⁰(1 + 2²²)

⇒ Clearly, N²+ 2^N is an integral multiple of 2¹⁰ which is in form of 2^x

∴ Largest possible value of x = 10