AMC 8 1998 Problem 10

AMC8 1998 年第 10 题

特殊数与不定方程★★☆☆☆

Each of the letters

\text{W}

\text{X}

\text{Y}

, and

\text{Z}

represents a different integer in the set

\{ 1,2,3,4\}

, but not necessarily in that order. If

\dfrac{\text{W}}{\text{X}} - \dfrac{\text{Y}}{\text{Z}}=1

, then the sum of

\text{W}

and

\text{Y}

字母

\text{W}

、

\text{X}

、

\text{Y}

和

\text{Z}

各代表集合

\{1,2,3,4\}

中不同的整数，但不一定按此顺序。若

\dfrac{\text{W}}{\text{X}} - \dfrac{\text{Y}}{\text{Z}}=1

，则

\text{W}

与

\text{Y}

之和为

正确答案 · Correct Answer

解析 · Solution

Solution 1

There are different ways to approach this problem, and I'll start with the different factor of the numbers of the set

\{ 1,2,3,4\}

1 has factor 1.

2 has factors 1 and 2. 3 has factors 1 and 3. 4 has factors 1, 2, and 4.

From here, we note that even though all numbers have the factor 1, only 4 has another factor other than 1 in the set (ie. 2 )

We could therefore have one fraction be

\displaystyle \frac{4}{2}

and another

\displaystyle \frac{3}{1}

The sum of the numerators is 4+3=7=E

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