AMC10 2014 A

AMC10 2014 A · Q20

AMC10 2014 A · Q20. It mainly tests Digit properties (sum of digits, divisibility tests), Base representation.

The product $(8)(888\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of 1000. What is $k$?

乘积 $(8)(888\dots8)$，其中第二个因数有 $k$ 个数字，是一个各位数字之和为 1000 的整数。$k$ 是多少？

(A) 901 901

(B) 911 911

(D) 991 991

(E) 999 999

Answer

Correct choice: (D)

正确答案：（D）

Solution

By direct multiplication, $8 \cdot 888 \dots 8 = 7111 \dots 104$, where the product has 2 fewer ones than the number of digits in $888 \dots 8$. Because $7 + 4 = 11$, the product must have $1000 - 11 = 989$ ones, so $k - 2 = 989$ and $k = 991$.

通过直接乘法，$8 \cdot 888 \dots 8 = 7111 \dots 104$，其中乘积中的 1 的个数比 $888 \dots 8$ 的位数少 2 个。因为 $7 + 4 = 11$，乘积中必须有 $1000 - 11 = 989$ 个 1，故 $k - 2 = 989$，$k = 991$。

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