We have : BCNN (2;3;5) = 30

=> BC (2 , 3 ,5) =  \(\left\{30;60;90;120;...\right\}\)

These numbers are less than 1000

=> BC ( 2, 3 , 5) < 1000 =  \(\left\{30;60;90;120;...;990\right\}\) (1)

Rally (1) has it all : 

\(\dfrac{990-30}{30}\) +1 = 33 (numbers)

Other way , in rally (1) the numbers are B (8) = \(\left\{120;240;...;960\right\}\)  (2)

Rally (2) has it all : 

\(\dfrac{960-120}{120}\) = 8 (numbers)

So has all positive integers in the set of numbers from 1 to 1000 :

33 - 8 = 25 ( numbers) is a multiple of 2, 3 and 5 but not 8