2008 AIME II Problems/Problem 8
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Problem
Let . Find the smallest positive integer such that is an integer.
Solution
By the product-to-sum identities, we have that . Therefore, this reduces to a telescoping series:
Thus, we need to be an integer; this can be only , which occur when is an integer. Thus . It easily follows that is the smallest such integer.
See also
2008 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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