2019 AMC 10B Problems/Problem 12

Problem

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

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

Solution

Convert $2019$ to base $7$ . This will get you $5613_7$ , which will be the upper bound. To maximize the sum of the digits, we want as many $6$ s as possible (which is the highest value in base $7$ ), and this would be the number $4666_7$ . Thus, the answer is $4+6+6+6 = \boxed{C) 22}$

Note: the number can also be $5566_7$ , which will also give the answer of $22$ .

iron Edited by greersc

2019 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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2019 AMC 10B Problems/Problem 12

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