2024 AMC 8 Problems/Problem 5: Difference between revisions

Latest revision as of 20:59, 20 August 2025

Problem

Aaliyah rolls two standard 6-sided dice. She notices that the product of the two numbers rolled is a multiple of $6$ . Which of the following integers cannot be the sum of the two numbers?

$\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 7\qquad\textbf{(D) } 8\qquad\textbf{(E) } 9$

Solution 1

First, figure out all pairs of numbers whose product is 6. Then, using the process of elimination, we can find the following:

$\textbf{(A)}$ is possible: $2\times 3$

$\textbf{(C)}$ is possible: $1\times 6$

$\textbf{(D)}$ is possible: $2\times 6$

The only integer that cannot be the sum is $\boxed{\textbf{(B) } 6}$ .

Solution 2

First, see that in order for the two numbers to have a product of 6, it must either have 6 as a factor or have factors that have a product of 6 (Notably $2\times 3$ ).

When we look at the answer choices, we see that answers $C, D, E$ all are more than 6, so they can all be added as a sum of 6. Answer $A$ works because $2+3=5$ , so 6 is the only answer choice that doesn't work. Thus, the answer is $\boxed{\textbf{(B) } 6}$ .

Video Solution 1 (Detailed Explanation) 🚀⚡📊

Youtube Link ⬇️

https://youtu.be/SreOL9LLmiI

~ ChillGuyDoesMath :)

Video Solution by Central Valley Math Circle (Goes through full thought process)

https://youtu.be/SjMK0jrVgj0

~mr_mathman

Video by MathTalks_Now

https://www.youtube.com/watch?v=crn37TRMLv4

-rc1219

Video Solution by Math-X (MATH-X)

https://youtu.be/BaE00H2SHQM?si=W8N4vwOx84KauOA6&t=1108

~Math-X

Video Solution (A Clever Explanation You’ll Get Instantly)

https://youtu.be/5ZIFnqymdDQ?si=eqTGEN2doXnn-oZE&t=443

~hsnacademy

Video Solution (easy to digest) by Power Solve

https://youtu.be/HE7JjZQ6xCk?si=2M33-oRy1ExY3_6l&t=239

Video Solution by NiuniuMaths (Easy to understand!)

https://www.youtube.com/watch?v=Ylw-kJkSpq8

~NiuniuMaths

Video Solution 2 by SpreadTheMathLove

https://www.youtube.com/watch?v=L83DxusGkSY

Video Solution by CosineMethod [🔥Fast and Easy🔥]

https://www.youtube.com/watch?v=51pqs80PUnY

Video Solution by Interstigation

https://youtu.be/ktzijuZtDas&t=296

Video Solution by Daily Dose of Math (Understandable, Quick, and Speedy)

https://youtu.be/bSPWqeNO11M?si=HIzlxPjMfvGM5lxR

~Thesmartgreekmathdude

Video Solution by WhyMath

https://youtu.be/QD016SsgJBQ

Video solution by TheNeuralMathAcademy

https://youtu.be/f63MY1T2MgI&t=346s

@@ Line 14: / Line 14: @@
 <math>\textbf{(D)}</math> is possible: <math>2\times 6</math>
-<math>\textbf{(E)}</math> is possible: <math>3\times 6</math>
+The only integer that cannot be the sum is <math>\boxed{\textbf{(B) } 6}</math>.
-The only integer that cannot be the sum is <math>\boxed{\textbf{(B)\ 6}}.</math>
+==Solution 2==
--ILoveMath31415926535 & countmath1 & Nivaar
+First, see that in order for the two numbers to have a product of 6, it must either have 6 as a factor or have factors that have a product of 6 (Notably <math>2\times 3</math>).
-==Video Solution 1 (easy to digest) by Power Solve==
+When we look at the answer choices, we see that answers <math>C, D, E</math> all are more than 6, so they can all be added as a sum of 6. Answer <math>A</math> works because <math>2+3=5</math>, so 6 is the only answer choice that doesn't work. Thus, the answer is <math>\boxed{\textbf{(B) } 6}</math>.
-https://youtu.be/HE7JjZQ6xCk?si=2M33-oRy1ExY3_6l&t=239
+== Video Solution 1 (Detailed Explanation) 🚀⚡📊 ==
+Youtube Link ⬇️
+https://youtu.be/SreOL9LLmiI
+~ ChillGuyDoesMath :)
+==Video Solution by Central Valley Math Circle (Goes through full thought process)==
+https://youtu.be/SjMK0jrVgj0
+~mr_mathman
+== Video by MathTalks_Now ==
+https://www.youtube.com/watch?v=crn37TRMLv4
+-rc1219
-==Video Solution by Math-X (First fully understand the problem!!!)==
+==Video Solution by Math-X (MATH-X)==
 https://youtu.be/BaE00H2SHQM?si=W8N4vwOx84KauOA6&t=1108
 ~Math-X
+==Video Solution (A Clever Explanation You’ll Get Instantly)==
+https://youtu.be/5ZIFnqymdDQ?si=eqTGEN2doXnn-oZE&t=443
+~hsnacademy
+==Video Solution  (easy to digest) by Power Solve==
+https://youtu.be/HE7JjZQ6xCk?si=2M33-oRy1ExY3_6l&t=239
 ==Video Solution by NiuniuMaths (Easy to understand!)==
@@ Line 41: / Line 69: @@
 ==Video Solution by Interstigation==
 https://youtu.be/ktzijuZtDas&t=296
+==Video Solution by Daily Dose of Math (Understandable, Quick, and Speedy)==
+https://youtu.be/bSPWqeNO11M?si=HIzlxPjMfvGM5lxR
+~Thesmartgreekmathdude
+==Video Solution by WhyMath==
+https://youtu.be/QD016SsgJBQ
+== Video solution by TheNeuralMathAcademy ==
+https://youtu.be/f63MY1T2MgI&t=346s
 ==See Also==
 {{AMC8 box|year=2024|num-b=4|num-a=6}}
 {{MAA Notice}}
+[[Category:Introductory Combinatorics Problems]]

2024 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
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
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