How Many unique ways are there to form this set A?

N is the set of all Natural Numbers from 1 till 500.
N is the set of all Natural Numbers from 1 till 500.

From this set we have form a set A such that

1. Any pair of 2 elements of A are relatively prime that is if x,y element of A then HCF(x,y)=1

2. A has got Maximum possible number of Elements

Now the question is

How many unique ways are there to form this set A such that Both the rules [1,2] are satisfied.

Solution:

5760

Explanation:

These are the following figures: 250*68*33*19*11*8*5*3
where 250 is for multiples of 2 (only one can be picked),
68 is for multiples of 3 less multiples of 6 (only one to pick),
33 for multiples of 5,
19 for 7,
11 for 11,
8 for 13,
5 for 17 and 3 for 19
replace 2 with powers of 2 of less than 500, so u have 8 options, simlarly 5 for 3 so the ans is
8*5*3*3*2*2*2*2 for 2,3,5,7,11,13,17 and 19 respectively

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