You have been asked to select a positive integer N which is less than 1000, such that it is either a multiple of 4, or a multiple of 6, or an odd multiple of 9. The number of such numbers is

There are $388$ integers. The multiples of $4$ or $6$ number $249+166-83=332$, with the multiples of $12$ subtracted once, and there are $56$ odd multiples of $9$ disjoint from that group. The sum $471$ double-counts the multiples of $12$.