Question
Without actually performing the long division, find if $\frac{987}{10500}$ will have terminating or non-terminating (repeating) decimal expansion. Give reasons for your answer.
✓
Answer
$ \frac{987}{10500}=\frac{3\times7\times47}{2^2\times3^1\times5^3\times7^1}=\frac{47}{2^2\times5^3}$
As denominator has prime factors only in 2 and 5 so number $\frac{987}{10500}$ is terminating decimal.
$\frac{47}{2^3\times5^3}\times2=\frac{94}{1000}=0.094$
$\begin{array}{c|c} 3 & 987 \\ \hline 7 & 329 \\ \hline & 47 \end{array}$ $\begin{array}{c|c} 5 & 10500 \\ \hline 3 & 2100 \\ \hline 7 & 700 \\ \hline 5 & 100 \\ \hline 5 & 20 \\ \hline 2 & 4 \\ \hline & 2 \end{array}$
As denominator has prime factors only in 2 and 5 so number $\frac{987}{10500}$ is terminating decimal.
$\frac{47}{2^3\times5^3}\times2=\frac{94}{1000}=0.094$
$\begin{array}{c|c} 3 & 987 \\ \hline 7 & 329 \\ \hline & 47 \end{array}$ $\begin{array}{c|c} 5 & 10500 \\ \hline 3 & 2100 \\ \hline 7 & 700 \\ \hline 5 & 100 \\ \hline 5 & 20 \\ \hline 2 & 4 \\ \hline & 2 \end{array}$
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