A number theory problem by Yusei Miyaji

Number Theory Level 2

Does 47 30 \frac{47}{30} have a recurring decimal?

Yes No

Yusei Miyaji by Yusei Miyaji , 17, Malaysia

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1 solution

Naren Bhandari
Oct 30, 2017

e 30 = 47 30 = 1 − 17 30 = 1 − 20 − 3 30 = 1 + 1 10 − 2 × 3 3 × 3 = 1 + 1 10 − 6 1 0 1 − 1 = 1.1 − recurring part \begin{aligned} \frac{e}{30} & = \frac{47}{30} \\& = 1-\frac{17}{30} \\& = 1- \frac{20-3}{30} \\& = 1+\frac{1}{10} -\frac{2\times 3}{3\times3} \\& = 1+\frac{1}{10} - {\color{#3D99F6}\frac{6}{10^1-1}} \\& = 1.1 - \text{recurring part}\end{aligned} The blue colored parts shows that the recurring will takes places of 6 with length of 1 1 after decimal point .Hence e 47 \frac{e}{47} is a recurring number.

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