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

Jmaster10 Feb 1, 2021

#1**0 **

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.

textot Feb 1, 2021