2014 AMC 12A Problem 16

Attempt Problem 16 of the 2014 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2014 AMC 12A solutions, or check the answer key.

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16.

The product (8)(8888),(8)(888\ldots8), where the second factor has kk digits, is an integer whose digits have a sum of 1000.1000. What is k?k?

901901

911911

919919

991991

999999

Answer: D
Concepts:digitspattern recognitionsmall cases
Difficulty rating: 1660
Solution:

By carrying out the multiplication, 8888k=711k204,8\cdot\underbrace{88\ldots8}_{k}=7\underbrace{1\ldots1}_{k-2}04, which has k2k-2 ones.

The digit sum is 7+(k2)+0+4=k+9.7+(k-2)+0+4=k+9. Setting k+9=1000k+9=1000 gives k=991.k=991.

Thus, the correct answer is D.

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