AMC 8 1998 Problem 22

AMC8 1998 年第 22 题

余数与同余★★★☆☆

Terri produces a sequence of positive integers by following three rules. She starts with a positive integer, then applies the appropriate rule to the result, and continues in this fashion. Rule 1: If the integer is less than 10, multiply it by 9. Rule 2: If the integer is even and greater than 9, divide it by 2. Rule 3: If the integer is odd and greater than 9, subtract 5 from it. A sample sequence:

23, 18, 9, 81, 76, \ldots .

Find the

98^\text{th}

term of the sequence that begins

98, 49, \ldots .

Terri 按照以下三条规则生成一个正整数序列。她从一个正整数开始，然后对结果应用适当的规则，并继续以此方式进行。

规则 1：如果整数小于 10，乘以 9。规则 2：如果整数为偶数且大于 9，除以 2。规则 3：如果整数为奇数且大于 9，减去 5。

示例序列：

23, 18, 9, 81, 76, \ldots

。

求以

98, 49, \ldots

开头的序列的第 98 项。

A.6
B.11
C.22
D.27
E.54

正确答案 · Correct Answer

解析 · Solution

Solution 1

We could start by looking for a pattern.

98, 49, 44, 22, 11, 6, 54, 27, 22, 11, 6, \ldots.

From here, we see that we have a pattern of

22, 11, 6, 54, 27, \ldots.

after 98, 49, 44

Our problem is now really

Find the 95-th term of the sequence that goes

22, 11, 6, 54, 27, 22, 11, 6, 54, 27, 22, \ldots.

There are 5 terms in each repetition of the pattern, and

95\equiv0\pmod{5}

, so the answer is

D

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