Difference between revisions of "2019 AMC 10B Problems/Problem 12"
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\qquad\textbf{(E) } 27</math> | \qquad\textbf{(E) } 27</math> | ||
− | ==Solution== | + | ==Solution 1== |
− | + | Observe that <math>2019_{10} = 5613_7</math>. To maximize the sum of the digits, we want as many <math>6</math>s as possible (since <math>6</math> is the highest value in base <math>7</math>), and this will occur with either of the numbers <math>4666_7</math> or <math>5566_7</math>. Thus, the answer is <math>4+6+6+6 = 5+5+6+6 = \boxed{\textbf{(C) }22}</math>. | |
− | + | ~IronicNinja went through this test 100 times | |
==Solution 2== | ==Solution 2== | ||
− | Note that all base 7 numbers with 5 | + | Note that all base <math>7</math> numbers with <math>5</math> or more digits are in fact greater than <math>2019</math>. Since the first answer that is possible using a <math>4</math> digit number is <math>23</math>, we start with the smallest base <math>7</math> number that whose digits sum to <math>23</math>, namely <math>5666_7</math>. But this is greater than <math>2019_{10}</math>, so we continue by trying <math>4666_7</math>, which is less than 2019. So the answer is <math>\boxed{\textbf{(C) }22}</math>. |
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+ | LaTeX code fix by EthanYL | ||
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+ | ==Video Solution== | ||
+ | https://youtu.be/mXvetCMMzpU | ||
+ | |||
+ | ~IceMatrix | ||
==See Also== | ==See Also== | ||
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{{AMC10 box|year=2019|ab=B|num-b=11|num-a=13}} | {{AMC10 box|year=2019|ab=B|num-b=11|num-a=13}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
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Latest revision as of 04:10, 28 January 2020
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
Video Solution
~IceMatrix
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.