Question

How many natural numbers n are there such that n! + 10 is a perfect square?

(a) 1

(b) 2

(c) 4

(d) infinitely many

Answer 1

Answer is : (a) 1

Given, n! + 10

Put n = 1, 2, 4, 5

n! + 10 is not a perfect square.

Put n = 3, 3! + 10 = 6+10 = 16 is a perfect square.

If n > 5

n! is multiple of 10.

∴ Product of odd and even is not a perfect square.