2019 AMC 10B Problems/Problem 12
Contents
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
~Education, the Study of Everything
Video Solution
See Also
| 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 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
