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This problem is based on decimal system conversion. In the given problem we have to convert the given number to the another number system from decimal and finding the sum of the digits later on.

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

$\textbf{(A) } 11 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 22 \qquad\textbf{(D) } 23 \qquad\textbf{(E) } 27$

Source

Competency

Difficulty

Suggested Book

AMC 10B, 2019 Problem 12

Decimal system conversion

6 out of 10

challenges and thrills of pre college mathematics

First hint

First of all 2019 is in base 10, see below

\(2019=2*10^3+0*10^2+1*10^1+9+10^0\).

so we have to convert it in the system having base 7.

Second Hint

After converting we will get

But we have to maximize the sum of digits. So we need to increase the number of 6 in the converted number(in base 7) and it is because 6 is the largest number in the number system having base 7.

Final Step

The conversion of maximize 6 in the number will occur with either of the numbers $4666_7$ or $5566_7$.

and now we can simply find the sum of the digits.

- https://www.cheenta.com/problem-related-to-triangle-amc-10b-2019-problem-10/
- https://www.youtube.com/watch?v=FNXm1dKvZ4I

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