Find the sum of the digits in the answer to underbrace9999cdots 99_94text nines × underbrace4444cdots 44_94text fours where a string of 94 nines is multiplied by a string of 94 fours.

AMC 8 1994 Problem 25

AMC8 1994 年第 25 题

数列与规律★★★★☆

Find the sum of the digits in the answer to

\underbrace{9999\cdots 99}_{94\text{ nines}} \times \underbrace{4444\cdots 44}_{94\text{ fours}}

where a string of 94 nines is multiplied by a string of 94 fours.

求以下乘积的各位数字之和：

\underbrace{9999\cdots 99}_{94\text{ 个 9}} \times \underbrace{4444\cdots 44}_{94\text{ 个 4}}

即 94 个 9 组成的数乘以 94 个 4 组成的数。

正确答案 · Correct Answer

解法1

常规解法 · 找规律观察

9\cdot4=36

（数字和 9）、

99\cdot44=4356

（数字和 18）；一般地

n

个 9 乘

n

个 4 的数字和为

9n

。故答案

9\cdot94=846

。

Solution 1

Elementary · find the pattern Notice the pattern:

9\cdot4=36

has digit sum 9, and

99\cdot44=4356

has digit sum 18; in general a string of

n

nines times

n

fours has digit sum

9n

. So the answer is

9\cdot94=846

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