Factorial Funda!

We can see that for all primes $n$ , $(n-1)!$ is not divisible by $n$ .As there are 168 primes less than 1000,there are at most 832 possibilities for $n$ .However, $n = 4$ does not satisfy the conditions.It is easy to verify that for all $n \geq 5$ where $n$ is not a prime, $n$ satisfies the conditions.

From 1 to 1000 there are 168 primes(this assumption put because only 1 and primes it self who can divide) and only 4 which is not primes who can't be divided by 3!, so there are 1000-168-1=831 natural numbers

Subtract all the prime no from 1000 including 4 so the answer will be 1000 -169=831

$\frac{0!}{1}$ = $\frac{1}{1}$ [satisfies the given condition]

$\frac{1!}{2}$ = $\frac{1}{2}$ [doesn't satisfies the given condition]

$\frac{2!}{3}$ = $\frac{2}{3}$ [doesn't satisfies the given condition]

$\frac{3!}{4}$ = $\frac{6}{4}$ [doesn't satisfies the given condition]

$\frac{4!}{5}$ = $\frac{24}{5}$ [doesn't satisfies the given condition]

$\frac{5!}{6}$ = $\frac{120}{6}$ [satisfies the given condition]

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You will notice that when n = all primes including 4 don't satisfies the given condition...

http://www.infoplease.com/ipa/A0001737.html

Go to the above link. You will get to know that there are 168 primes between 1 to 100.

Then add 1 to 168.

Since $\frac{3!}{4}$ does not satisfy this condition.

Thus the answer is 1000-169 = $\boxed{831}$

Note-It is very difficult if you go on trying trial and error method for finding primes!!