Difference between revisions of "2024 AMC 10A Problems/Problem 3"

Latest revision as of 19:18, 26 May 2025

Problem

What is the sum of the digits of the smallest prime that can be written as a sum of $5$ distinct primes?

$\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }10\qquad\textbf{(E) }13$

Solution 1

Let the requested sum be $S.$ Recall that $2$ is the only even (and the smallest) prime, so $S$ is odd. It follows that the five distinct primes are all odd. The first few odd primes are $3,5,7,11,13,17,19,\ldots.$ We conclude that $S>3+5+7+11+13=39,$ as $39$ is a composite. The next possible value of $S$ is $3+5+7+11+17=43,$ which is a prime. Therefore, we have $S=43,$ and the sum of its digits is $4+3=\boxed{\textbf{(B) }7}.$

~MRENTHUSIASM

Solution 2

We notice that the minimum possible value of the sum of $5$ odd distinct primes is $3 + 5 + 7 + 11 + 13 = 39$ , which is not a prime. The smallest prime greater than that is $41$ . However, this cannot be written as the sum of $5$ distinct primes, since $15$ is not prime. However, $43$ can be written as $3 + 5 + 7 + 11 + 17 = 43$ , so the answer is $4 + 3 = \boxed{\textbf{(B) }7}$

~andliu766

Video Solution

https://youtu.be/l3VrUsZkv8I ~MC

Video Solution by Central Valley Math Circle

https://youtu.be/VJpw7eu7s6M

~mr_mathman

Video Solution by Daily Dose of Math

https://youtu.be/4mf18UuZENw

~Thesmartgreekmathdude

Video Solution 1 by Power Solve

https://youtu.be/j-37jvqzhrg?si=5uruPdMajz7B8jhy&t=307

Video Solution 2 by SpreadTheMathLove

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

Video Solution by TheBeautyofMath

For AMC 10: https://youtu.be/uKXSZyrIOeU?t=900

~IceMatrix

Video Solution by Dr. David

https://youtu.be/Gka4ivb0AEw

@@ Line 1: / Line 1: @@
-everyone gangsta until the real problems come out.
+== Problem ==
+What is the sum of the digits of the smallest prime that can be written as a sum of <math>5</math> distinct primes?
+<math>\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }10\qquad\textbf{(E) }13</math>
+== Solution 1==
+Let the requested sum be <math>S.</math> Recall that <math>2</math> is the only even (and the smallest) prime, so <math>S</math> is odd. It follows that the five distinct primes are all odd. The first few odd primes are <math>3,5,7,11,13,17,19,\ldots.</math> We conclude that <math>S>3+5+7+11+13=39,</math> as <math>39</math> is a composite. The next possible value of <math>S</math> is <math>3+5+7+11+17=43,</math> which is a prime. Therefore, we have <math>S=43,</math> and the sum of its digits is <math>4+3=\boxed{\textbf{(B) }7}.</math>
+~MRENTHUSIASM
+== Solution 2 ==
+We notice that the minimum possible value of the sum of <math>5</math> odd distinct primes is <math>3 + 5 + 7 + 11 + 13 = 39</math>, which is not a prime. The smallest prime greater than that is <math>41</math>. However, this cannot be written as the sum of <math>5</math> distinct primes, since <math>15</math> is not prime. However, <math>43</math> can be written as <math>3 + 5 + 7 + 11 + 17 = 43</math>, so the answer is <math>4 + 3 = \boxed{\textbf{(B) }7}</math>
+~andliu766
+==Video Solution==
+https://youtu.be/l3VrUsZkv8I
+~MC
+==Video Solution by Central Valley Math Circle==
+https://youtu.be/VJpw7eu7s6M
+~mr_mathman
+== Video Solution by Daily Dose of Math ==
+https://youtu.be/4mf18UuZENw
+~Thesmartgreekmathdude
+== Video Solution 1 by Power Solve ==
+https://youtu.be/j-37jvqzhrg?si=5uruPdMajz7B8jhy&t=307
+==Video Solution 2 by SpreadTheMathLove==
+https://www.youtube.com/watch?v=6SQ74nt3ynw
+==Video Solution by TheBeautyofMath==
+For AMC 10: https://youtu.be/uKXSZyrIOeU?t=900
+~IceMatrix
+==Video Solution by Dr. David==
+https://youtu.be/Gka4ivb0AEw
+==See also==
+{{AMC10 box|year=2024|ab=A|num-b=2|num-a=4}}
+{{MAA Notice}}

2024 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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