What-is-the-smallest-positive-integer-x-for-which-1-32-x-10-y-for-some-positive-integer-y-

Question Number 11606 by Joel576 last updated on 29/Mar/17

What is the smallest positive integer x for

which \frac{1}{32} = \frac{x}{10^{y}} for some positive integer y ?

Answered by mrW1 last updated on 29/Mar/17

32 x = 10^{y} = {(2 \times 5)}^{y} = 2^{y} \times 5^{y}

2^{5} x = 2^{y} \times 5^{y}

x = 2^{y - 5} \times 5^{y}

f o r x t o b e p o s i t i v e i n t e g e r, i t m u s t b e

y - 5 ⩾ 0 o r y ⩾ 5

\Rightarrow m i n . x = 2^{0} \times 5^{5} = 3125

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