How many positive integers , have the property that when is written as a decimal, the expansion eventually terminates?
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For n 1 to have a terminating decimal, the number n 1 0 k must be an integer for some value of k (specifically, 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 . In the table below, we summarize all the values of a , b which will give n < 1 0 0 0 .
a 4 3 2 1 0 b 0 0 , 1 , 2 0 , … , 5 0 , … , 7 0 , … , 9
This gives a total of 1 + 3 + 6 + 8 + 1 0 = 2 8 values for n .