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 ?
Solution 1
Observe that . To maximize the sum of the digits, we want as many
s as possible (since
is the highest value in base
), and this will occur with either of the numbers
or
. Thus, the answer is
.
~IronicNinja went through this test 100 times
Solution 2
Note that all base numbers with
or more digits are in fact greater than
. Since the first answer that is possible using a
digit number is
, we start with the smallest base
number that whose digits sum to
, namely
. But this is greater than
, so we continue by trying
, which is less than 2019. So the answer is
.
LaTeX code fix by EthanYL
Solution 3
Again note that you want to maximize the number of s to get the maximum sum. Note that
, so you have room to add a thousands digit base
. Fix the
in place and try different thousands digits, to get
as the number with the maximum sum of digits. The answer is
.
~mwu2010
Video Solution
~IceMatrix
See Also
2019 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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All AMC 10 Problems and Solutions |
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