Consider a set A = {1, 2, 3, …….., 1000}. How many members of A shall be…

Solution:

Count how many integers from 1 to 1000 are divisible by 3 or 5 using inclusion–exclusion.

• Numbers divisible by 3: floor(1000 ÷ 3) = 333.

• Numbers divisible by 5: floor(1000 ÷ 5) = 200.

• Numbers divisible by both 3 and 5 (i.e. by 15): floor(1000 ÷ 15) = 66.

• Apply inclusion–exclusion: 333 + 200 − 66 = 467.

Answer: There are 467 members of the set {1, …, 1000} divisible by 3 or 5 (or both).