Many calculators have a reciprocal key boxed(1/x) that replaces the current number displayed with its reciprocal. For example, if the display is boxed00004 and the boxed(1/x) key is depressed, then the display becomes boxed000.25 . If boxed00032 is currently displayed, what is the fewest number of times you must depress the boxed(1/x) key so the display again reads boxed00032 ?

AMC 8 1989 Problem 18

AMC8 1989 年第 18 题

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Many calculators have a reciprocal key

\displaystyle \boxed{\frac{1}{x}}

that replaces the current number displayed with its reciprocal. For example, if the display is

\boxed{00004}

and the

\displaystyle \boxed{\frac{1}{x}}

key is depressed, then the display becomes

\boxed{000.25}

. If

\boxed{00032}

is currently displayed, what is the fewest number of times you must depress the

\displaystyle \boxed{\frac{1}{x}}

key so the display again reads

\boxed{00032}

许多计算器有一个倒数键

\displaystyle \boxed{\frac{1}{x}}

，将当前显示的数替换为其倒数。例如，如果显示为

\boxed{00004}

并按下

\displaystyle \boxed{\frac{1}{x}}

键，则显示变为

\boxed{000.25}

。如果当前显示

\boxed{00032}

，最少需要按多少次

\displaystyle \boxed{\frac{1}{x}}

键才能使显示再次变为

\boxed{00032}

？

正确答案 · Correct Answer

Solution 1

After pressing the reciprocal key once, the calculator will display the decimal representation of

\displaystyle \frac{1}{32}

(which is 0.03125 ). After pressing the reciprocal key again, the calculator will display

\displaystyle 1 \div \frac{1}{32} = 32

So, after 2 presses the calculator will display 32 again. (Since

\displaystyle \frac{1}{32} = 0.03125

is a terminating decimal, there will not be any rounding.)

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