Problem:
How many whole numbers from through are divisible by either or or both?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
A number is divisible by if it is a multiple of , and it is divisible by if it is a multiple of . There are multiples of , and multiples of which are whole numbers less than . However, numbers and which are divisible by both and have been counted twice. Thus the total number which are divisible by either or or both is .
Using the Sieve of Eratosthenes and marking each number, multiples of , and each number, multiples of , yields
Thus, the total number which are divisible by either or or both is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions