Problem:
Let and be two arithmetic progressions. The set is the union of the first terms of each sequence. How many distinct numbers are in
Answer Choices:
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Solution:
The smallest number that appears in both sequences is . The two sequences have common differences and , whose least common multiple is , so a number appears in both sequences if and only if it is in the form
where is a nonnegative integer. Such a number is in the first terms of both sequences if and only if
Thus , so there are duplicate numbers. Therefore the number of distinct numbers is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions