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# For how many integer values of n between 1 and 120 inclusive does the decimal representation of n/120 terminate?

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For how many integer values of n between 1 and 120 inclusive does the decimal representation of n/120 terminate?

Jun 5, 2018

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For how many integer values of n between 1 and 120 inclusive does the decimal representation of n/120 terminate?

The terminating decimal is if and only if the denominator has powers of only 2 and/or 5

The denominator is $$120 = 2^3\cdot {\color{red}3} \cdot 5^1$$

So $$n$$ must drop the 3 in the denominator.

$$n = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \ldots , 117, 120$$

$$\begin{array}{|lrcll|} \hline AP: & a_n &=& 3+(m-1)*3 \quad & | \quad a_n = 120 \\ & 120 &=& 3+(m-1) * 3 \\ & 120 &=& 3+3m - 3 \\ & 120 &=& 3m \\ & m &=& \frac{120}{3} \\ & \mathbf{m} & \mathbf{=} & \mathbf{40} \\ \hline \end{array}$$

40 integer values of n between 1 and 120 inclusive does the decimal representation of $$\frac{n}{120}$$ terminate.

Jun 5, 2018