My generalization to a problem , not sure it is true

I am doing the past Putnam problems and one is

Prove that is an integer .

I make a generalization , which is :

is an integer iff , where is the minimun prime factor of .

If it is true , then the problem from Putnam is just a special case . However , I couldn't trust my proof so i beg for help here .

Thanks

p.s. I make use of floor function to prove it .

EDIT: I find some errors in my statement , iff should be changed to if and the equality can hold . i.e.