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HMMT November, 2013

Number theory Difficulty 4.9 AIME Prove it United States

Problem:
Find the sum of 1n\frac{1}{n} over all positive integers nn with the property that the decimal representation of 1n\frac{1}{n} terminates.

Solution

Solution:
The decimal representation of 1n\frac{1}{n} terminates if and only if n=2i5jn = 2^{i} 5^{j} for some nonnegative integers i,ji, j, so our desired sum is
∑i≥0∑j≥02−i5−j=∑i≥02−i∑j≥05−j=(1−2−1)−1(1−5−1)−1=2154=52 \sum_{i \geq 0} \sum_{j \geq 0} 2^{-i} 5^{-j} = \sum_{i \geq 0} 2^{-i} \sum_{j \geq 0} 5^{-j} = \left(1 - 2^{-1}\right)^{-1} \left(1 - 5^{-1}\right)^{-1} = \frac{2}{1} \frac{5}{4} = \frac{5}{2}

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