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For how many integer values of n between 1 and 1000 inclusive does the decimal representation of n/1375 + n/3 terminate?

 Feb 12, 2021
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Note that 1400 = 2^3 * 5^2 * 7.

 

A fraction (reduced to lowest terms) has a terminating expansion if and only if the only prime factors in the denominator are 2 and 5.

In the case of n/1400, we therefore need n to be divisible by 7.

Hence, there are floor(1000/7) = 142 possible values of n.

 

 

I hope this helps!

 Feb 12, 2021

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