Hello,
. Consider two cases :
is odd. Let and . , and . Since , .
is even. Let and , for some odd prime. , and . Since , if and only if . Since a number cannot potentially divide every prime less than itself (otherwise it would at least be the product of all these primes and would be much greater, leaving more primes in between), there must exist some for which , and follows.
As you can see, the case for even is much harder than the case for odd. Can you see how I used divisibility theorems and properties of prime numbers to solve this problem ? If you have any questions/concerns, feel free to ask.