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| {{duplicate|[[2024 AMC 10A Problems/Problem 4|2024 AMC 10A #4]] and [[2024 AMC 12A Problems/Problem 3|2024 AMC 12A #3]]}}
| | #redirect [[2024 AMC 12A Problems/Problem 3]] |
| == Problem ==
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| The number <math>2024</math> is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?
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| <math>\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24</math>
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| == Solution 1 ==
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| Since we want the least number of two-digit numbers, we maximize the two-digit numbers by choosing as many <math>99</math>s as possible. Since <math>2024=99\cdot20+44\cdot1,</math> we choose twenty <math>99</math>s and one <math>44,</math> for a total of <math>\boxed{\textbf{(B) }21}</math> two-digit numbers.
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| ~MRENTHUSIASM
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| == Solution 2 ==
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| We claim the answer is <math>21</math>. This can be achieved by adding twenty <math>99</math>'s and a <math>44</math>. To prove that the answer cannot be less than or equal to <math>20</math>, we note that the maximum value of the sum of <math>20</math> or less two digit numbers is <math>20 \cdot 99 = 1980</math>, which is smaller than <math>2024</math>, so we are done. Thus, the answer is <math>\boxed{\textbf{(B) }21}</math>.
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| ~andliu766
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| == Solution 3 (Same as Solution 1 but Using 100=99+1)==
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| <math>2024=100\cdot20+24</math>. Since <math>100=99+1</math>, <math>2024=(99+1)\cdot20+24=99\cdot20+1\cdot20+24=99\cdot20+44</math>. Therefore a total of <math>\boxed{\textbf{(B) }21}</math> two-digit numbers are needed.
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| ~woh123
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| == Solution 4 ==
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| The maximum <math>2</math>-digit number is <math>99</math>, but try <math>100</math>. <math>2024 \div 100</math> is a little more than <math>20</math>, and the remainder is less than <math>100</math>, by intuition, so there's <math>20 +</math> the remainder <math>1 = \boxed{\textbf{(B) }21}</math>.
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| ~RandomMathGuy500
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| == Video Solution by Math from my desk ==
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| https://www.youtube.com/watch?v=f6ogWpv56qw
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| == Video Solution (⚡️ 55 sec solve ⚡️) ==
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| https://youtu.be/5nsOZQTcyMs
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| <i>~Education, the Study of Everything </i>
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| == Video Solution by Pi Academy ==
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| https://youtu.be/GPoTfGAf8bc?si=JYDhLVzfHUbXa3DW
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| == Video Solution by Daily Dose of Math ==
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| https://youtu.be/sEk9jQnMzfk
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| ~Thesmartgreekmathdude
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| == Video Solution by FrankTutor ==
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| https://youtu.be/g2RxRsxrp2Y
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| == Video Solution 1 by Power Solve ==
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| https://youtu.be/j-37jvqzhrg?si=rWQoAYu7QsZP8ty4&t=407
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| ==Video Solution by SpreadTheMathLove==
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| https://www.youtube.com/watch?v=6SQ74nt3ynw
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| ==See also==
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| {{AMC10 box|year=2024|ab=A|num-b=3|num-a=5}}
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| {{AMC12 box|year=2024|ab=A|num-b=2|num-a=4}}
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| {{MAA Notice}}
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