5 Times Larger

Number Theory Level 2

Find the least number whose last digit is 7 and which becomes 5 times larger when this last digit is carried to the beginning of the number.


The answer is 142857.

Nilotpal Chakraborty by Nilotpal Chakraborty , 21, India

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1 solution

Shashvat Shukla
Apr 13, 2014

Let x \Large x be the answer. Let 0. x ‾ = a b \large0.\overline{x}=\frac{a}{b} for some integers a and b. From the given info, 5 × a b = 0.7 x ‾ \large5\times\frac{a}{b}=0.7\overline{x} So, 50 × a b = 7. x ‾ = 7 + a b \large50\times\frac{a}{b}=7.\overline{x}=7+\frac{a}{b} Thus, 49 × a b = 7 ⇒ 0. x ‾ = a b = 1 7 = 0. 142857 ‾ \large49\times\frac{a}{b}=7\Rightarrow0.\overline{x}=\frac{a}{b}=\frac{1}{7}=0.\overline{142857} Hence x = 142857 \large x=142857

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The Best Solution I have EVER seen

Arijit ghosh Dastidar - 5 years, 12 months ago

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