2024 AMC 10A Problems/Problem 5: Difference between revisions

Latest revision as of 16:15, 18 August 2025

The following problem is from both the 2024 AMC 10A #5 and 2024 AMC 12A #4, so both problems redirect to this page.

Problem

What is the least value of $n$ such that $n!$ is a multiple of $2024$ ?

$\textbf{(A) } 11\qquad\textbf{(B) } 21\qquad\textbf{(C) } 22\qquad\textbf{(D) } 23\qquad\textbf{(E) } 253$

Solution

Note that $2024=2^3\cdot11\cdot23$ in the prime factorization. Since $23!$ is a multiple of $2^3, 11,$ and $23,$ we conclude that $23!$ is a multiple of $2024.$ Therefore, we have $n=\boxed{\textbf{(D) } 23}.$

Remark

Memorizing the prime factorization of the current year is useful for the AMC 8/10/12 Exams.

~MRENTHUSIASM

Remark

Note that $2024$ is $2025 -1$ , which is $45^2 -1 = (45+1)(45-1) = 46\cdot44$ .

~WISETIGERJ2

Video Solution by Central Valley Math Circle

https://youtu.be/Hc5DxRT-DOU

~mr_mathman

Video Solution

https://youtu.be/l3VrUsZkv8I

Video Solution by Math from my desk

https://www.youtube.com/watch?v=fAitluI5SoY&t=3s

Video Solution (⚡️ 1 min solve ⚡️)

https://youtu.be/FD6rV3wGQ74

~Education, the Study of Everything

Video Solution by Pi Academy

https://youtu.be/GPoTfGAf8bc?si=JYDhLVzfHUbXa3DW

Video Solution by Daily Dose of Math

https://youtu.be/DXDJUCVX3yU

~Thesmartgreekmathdude

Video Solution 1 by Power Solve

https://youtu.be/j-37jvqzhrg?si=qwyiAvKLbySyDR7D&t=529

Video Solution by SpreadTheMathLove

https://www.youtube.com/watch?v=6SQ74nt3ynw

Video Solution by TheBeautyofMath

For AMC 10: https://youtu.be/uKXSZyrIOeU?t=1168

For AMC 12: https://youtu.be/zaswZfIEibA?t=984

~IceMatrix

Video Solution by Dr. David

https://youtu.be/uhpWASWW2ns

Video solution by TheNeuralMathAcademy

https://www.youtube.com/watch?v=4b_YLnyegtw&t=782s

2024 AMC 10A (Problems • Answer Key • Resources)
Preceded by 2023 AMC 10B Problems	Followed by 2024 AMC 10B Problems
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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing

@@ Line 1: / Line 1: @@
-A user attempted to look at 2024 AMC 10A Problem 5. What is the probability that they can see the question before the actual test date?
+{{duplicate|[[2024 AMC 10A Problems/Problem 5|2024 AMC 10A #5]] and [[2024 AMC 12A Problems/Problem 4|2024 AMC 12A #4]]}}
+== Problem ==
+What is the least value of <math>n</math> such that <math>n!</math> is a multiple of <math>2024</math>?
-(A) 0% (B) 0% (C) 0% (D) 0% (E) 0% (F) skibidi
+<math>\textbf{(A) } 11\qquad\textbf{(B) } 21\qquad\textbf{(C) } 22\qquad\textbf{(D) } 23\qquad\textbf{(E) } 253</math>
+== Solution==
+Note that <math>2024=2^3\cdot11\cdot23</math> in the prime factorization. Since <math>23!</math> is a multiple of <math>2^3, 11,</math> and <math>23,</math> we conclude that <math>23!</math> is a multiple of <math>2024.</math> Therefore, we have <math>n=\boxed{\textbf{(D) } 23}.</math>
+<u><b>Remark</b></u>
+Memorizing the prime factorization of the current year is useful for the AMC 8/10/12 Exams.
+~MRENTHUSIASM
+<u><b>Remark</b></u>
+Note that <math>2024</math> is <math>2025 -1</math>, which is <math>45^2 -1 = (45+1)(45-1) = 46\cdot44</math>.
+~WISETIGERJ2
+==Video Solution by Central Valley Math Circle==
+https://youtu.be/Hc5DxRT-DOU
+~mr_mathman
+==Video Solution==
+https://youtu.be/l3VrUsZkv8I
+== Video Solution by Math from my desk ==
+https://www.youtube.com/watch?v=fAitluI5SoY&t=3s
+== Video Solution (⚡️ 1 min solve ⚡️) ==
+https://youtu.be/FD6rV3wGQ74
+<i>~Education, the Study of Everything</i>
+== Video Solution by Pi Academy ==
+https://youtu.be/GPoTfGAf8bc?si=JYDhLVzfHUbXa3DW
+== Video Solution by Daily Dose of Math ==
+https://youtu.be/DXDJUCVX3yU
+~Thesmartgreekmathdude
+== Video Solution 1 by Power Solve ==
+https://youtu.be/j-37jvqzhrg?si=qwyiAvKLbySyDR7D&t=529
+==Video Solution by SpreadTheMathLove==
+https://www.youtube.com/watch?v=6SQ74nt3ynw
+==Video Solution by TheBeautyofMath==
+For AMC 10: https://youtu.be/uKXSZyrIOeU?t=1168
+For AMC 12: https://youtu.be/zaswZfIEibA?t=984
+~IceMatrix
+==Video Solution by Dr. David==
+https://youtu.be/uhpWASWW2ns
+==Video solution by TheNeuralMathAcademy==
+https://www.youtube.com/watch?v=4b_YLnyegtw&t=782s
+==See Also==
+{{AMC10 box|year=2024|ab=A|before=[[2023 AMC 10B Problems]]|after=[[2024 AMC 10B Problems]]}}
+* [[AMC 10]]
+* [[AMC 10 Problems and Solutions]]
+* [[Mathematics competitions]]
+* [[Mathematics competition resources]]
+{{MAA Notice}}

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