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# terminating decimals

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125
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+161

For how many integer values of $$n$$ between 1 and 1000 inclusive does the decimal representation of $$\frac{n}{1375}$$ terminate?

Feb 1, 2021

#1
+354
+1

The only way a simplified fraction can terminate is if the denominator can be expressed as $$2^x \cdot 5^y$$, where x and y are integers because these are the factors of 10.

The prime factorization of 1375 is $$5^3 \cdot 11$$

Therefore, the only way to make this fraction a terminating decimal is if the numerator, n, is a multiple of 11, to cancel out the 11 in the denominator to make it terminate.

There are 90 numbers from 1 to 1000 which is a multiple of 11.

Therefore, the answer to this question is $$\boxed{90}$$ numbers that satisfy this condition.

Feb 1, 2021