2019 AMC 10 B

Complete problem set with solutions and individual problem pages

Problem 12 Easy

What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than 2019? (2019 AMC 10B Problem, Question#12)

  • A.

    11

  • B.

    14

  • C.

    22

  • D.

    23

  • E.

    27

Answer:C

Observe that 2019_{10}=5613_7.To maximize the sum of the digits, we want as many \rm 6s as possible (since 6 is the highest value in base 7), and this will occur with either of the numbers 4666_7 or 5566_7.

Thus, the answer is 4+6+6+6=\text {(C)}22..

\simIronicNinja, edited by some people,

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

Note that all base 7 numbers with 5 or more digits are in fact greater than 2019. Since the first answer that is possible using a 4 digit number is 23, we start with the smallest base 7 number that whose digits sum to 23, namely 5666_7, But this is greater than 2019_1 0, so we continue by trying 4666_7，

which is less than 2019. So the answer is \text {(C)}22..

Related Problems
Problem 10 2019 AMC 10 B
Problem 11 2019 AMC 10 B
Problem 13 2019 AMC 10 B
Problem 14 2019 AMC 10 B
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