2014 AMC 12A Problem 16

Problem 16 of 25IntermediateNumber Theory

The product

(8)(888\ldots8),

where the second factor has

k

digits, is an integer whose digits have a sum of

1000.

What is

k?

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Solution

By carrying out the multiplication,

8\cdot\underbrace{88\ldots8}_{k}=7\underbrace{1\ldots1}_{k-2}04,

which has

k-2

ones. The digit sum is

7+(k-2)+0+4=k+9.

Setting

k+9=1000

gives

k=991.

Thus, the correct answer is D.

Concepts: digits · pattern recognition · small cases

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