For each prime number p define Fp n in the natural numbers

For each prime number p, define F(p) = {n in the natural numbers | p is the smallest prime factor of n}. Show that {F(p)|p prime} partitions the set of natural numbers greater than 1.

Solution

Prime numbers = 2,3,5,7,11,13.....

F(p)= {n in natural number/smallest prime factor of n}={2/2,3/3,4/2.5/5,6/2,7/7,8/2.......}

F(p) = {1,1,2,1,3,1,4.....}

for p=2

F(p)=2/2=1

for p=4

F(p)=4/2

=2

for p=9

F(p)=3

and else

Thus for each example we can see that value of F(p)/p prime is always greater than 1