Problem:
Consider the following sets of elements each:
How many of these sets contain exactly two multiples of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Observe that 1000 divided by 7 is 142 with a remainder of 6 . There are therefore 142 positive multiples of 7 that are less than 1000 , namely . Because every set of 10 consecutive integers must contain at least one multiple of 7 and cannot contain three multiples of 7 , exactly of the given sets must contain exactly two multiples of 7 .
The problems on this page are the property of the MAA's American Mathematics Competitions