2019 AMC 10B Problem 12

Below is the professionally curated solution for Problem 12 of the 2019 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2019 AMC 10B solutions, or check the answer key.

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Concepts:number base digits optimization

Difficulty rating: 1430

12.

What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019?$

$11$

$14$

$22$

$23$

$27$

Solution:

First, $2019 = 5613_7.$ Therefore, the $4$ th digit from the left is at most $5.$ If that digit was $4$ and every other digit was maximized, then we get $4666_7,$ with a digit sum of $22.$ If that digit was $5,$ we can only have a sum greater than $22$ by making $5666_7$ which is too large.

Therefore, the largest digit sum is $22.$

Thus, the answer is C .

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