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Probability Level pending

How many positive integers n < 1000 n <1000 , have the property that when 1 n \frac{1}{n} is written as a decimal, the expansion eventually terminates?


The answer is 28.

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1 solution

Calvin Lin Staff
May 13, 2014

For 1 n \frac{1}{n} to have a terminating decimal, the number 1 0 k n \frac{10^k}{n} must be an integer for some value of k k (specifically, k k is the number of decimal places the number has.) Thus, a number will have this property if it has no prime factors aside from 2 and 5. Let n = 5 a 2 b . n = 5^a2^b. In the table below, we summarize all the values of a , b a,b which will give n < 1000. n < 1000.

a b 4 0 3 0 , 1 , 2 2 0 , , 5 1 0 , , 7 0 0 , , 9 \begin{array}{|r|l|} \hline a & b\\ \hline 4 & 0\\ 3 & 0,1,2\\ 2 & 0,\ldots,5\\ 1 & 0,\ldots,7\\ 0 & 0,\ldots,9\\ \hline \end{array}

This gives a total of 1 + 3 + 6 + 8 + 10 = 28 1 + 3 + 6 + 8 + 10 = 28 values for n . n.

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