How many natural numbers, n where n < 50 exist satisfying the condition that (n – 1)! is not a multiple of n?15161718

To solve this problem, we need to understand the properties of factorial and natural numbers.

The factorial of a number n, denoted as n!, is the product of all positive integers less than or equal to n. For example, 5! = 54321 = 120.

The question asks for natural numbers n, where n < 50, such that (n-1)! is not a multiple of n.

Let's consider the properties of factorials. For any natural number n, n! is always a multiple of n because n! = n*(n-1)(n-2)...32*1. Therefore, (n-1)! will not be a multiple of n for any n that is a prime number.

Prime numbers are natural numbers greater than 1 that have only two positive divisors: 1 and the number itself. The prime numbers less than 50 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.

However, there is one exception to this rule: the number 4. 4 is not a prime number, but 3! = 6, which is not a multiple of 4.

So, the natural numbers n < 50 for which (n-1)! is not a multiple of n are: 2, 3, 4, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.

There are 16 such numbers.