QUESTION :
- How many distinct-and-different { Category X Composite Numbers } can we formulate
- if we were to start-out with exactly [ Q ] prime-numbers ;
with the 2 special conditions being that :
- each-and-every-one of the prime-number/s must appear at-least-once as a prime-factor of the formulated-number [ N ] ,
and
- repetition of the prime-numbers as a prime-factor of the formulated-number [ N ] is allowed .
And we shall let the values of [ Q ] vary from [ 1 ] to [ 10 ] ,
- to arrive at 10 successive answers .
- [ A1 ] / [ A2 ] / [ A3 ] / [ A4 ] / [ A5 ] / [ A6 ] / [ A7 ] / [ A8 ] / [ A9 ] / [ A10 ] respectively
for the values of [ Q ] varying from [ 1 ] to [ 10 ] .
Then :
- the number of distinct-and-different { Category X Composite Numbers }
- that can be formulated with an initial set of [ H ] distinct-and-different prime-numbers
( assuming at this first-instance that the value of [ H ] is greater than [ 10 ] )
is then :
- that can be formulated with an initial set of [ H ] distinct-and-different prime-numbers
And we take special note here that :
- For the above Equation involving 10 terms :
- for the 1st term involving [ A1 ] ,
- each of the { Category X Composite Numbers } making-up the Count here
will have [ 1 ]-and-exactly-[ 1 ] prime-number as its prime-factors ;
- each of the { Category X Composite Numbers } making-up the Count here
- for the 2nd term involving [ A2 ] ,
- each of the { Category X Composite Numbers } making-up the Count here
will have [ 2 ]-and-exactly-[ 2 ] different prime-numbers to serve as its prime-factors ;
- each of the { Category X Composite Numbers } making-up the Count here
- for the 3rd term involving [ A3 ] ,
- each of the { Category X Composite Numbers } making-up the Count here
will have [ 3 ]-and-exactly-[ 3 ] different prime-numbers to serve as its prime-factors ;
- each of the { Category X Composite Numbers } making-up the Count here
- for the 4th term involving [ A4 ] ,
- each of the { Category X Composite Numbers } making-up the Count here
will have [ 4 ]-and-exactly-[ 4 ] different prime-numbers to serve as its prime-factors ;
- each of the { Category X Composite Numbers } making-up the Count here
- for the 5th term involving [ A5 ] ,
- each of the { Category X Composite Numbers } making-up the Count here
will have [ 5 ]-and-exactly-[ 5 ] different prime-numbers to serve as its prime-factors ;
- each of the { Category X Composite Numbers } making-up the Count here
- for the 6th term involving [ A6 ] ,
- each of the { Category X Composite Numbers } making-up the Count here
will have [ 6 ]-and-exactly-[ 6 ] different prime-numbers to serve as its prime-factors ;
- each of the { Category X Composite Numbers } making-up the Count here
- for the 7th term involving [ A7 ] ,
- each of the { Category X Composite Numbers } making-up the Count here
will have [ 7 ]-and-exactly-[ 7 ] different prime-numbers to serve as its prime-factors ;
- each of the { Category X Composite Numbers } making-up the Count here
- for the 8th term involving [ A8 ] ,
- each of the { Category X Composite Numbers } making-up the Count here
will have [ 8 ]-and-exactly-[ 8 ] different prime-numbers to serve as its prime-factors ;
- each of the { Category X Composite Numbers } making-up the Count here
- for the 9th term involving [ A9 ] ,
- each of the { Category X Composite Numbers } making-up the Count here
will have [ 9 ]-and-exactly-[ 9 ] different prime-numbers to serve as its prime-factors ;
- each of the { Category X Composite Numbers } making-up the Count here
- for the 10th-and-final term involving [ A10 ] ,
- each of the { Category X Composite Numbers } making-up the Count here
will have [ 10 ]-and-exactly-[ 10 ] different prime-numbers to serve as its prime-factors ;
- each of the { Category X Composite Numbers } making-up the Count here
- for the 1st term involving [ A1 ] ,
As such , the Equation we have above does give us the Full Count .