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

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== Problem ==
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https://www.linkedin.com/in/quandalious-dingleton-380a90243/
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3.I'm in the thick of it, everybody knows
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What is the sum of the digits of the smallest prime that can be written as a sum of <math>5</math> distinct primes?
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They know me where it snows, I skied in and they froze
 +
I don't know no nothin' 'bout no ice, I'm just cold
 +
Forty somethin' milli' subs or so, I've been told
 +
I'm in my prime and this ain't even final form
 +
They knocked me down, but still, my feet, they find the floor
 +
I went from living rooms straight out to sold-out tours
 +
Life's a fight, but trust, I'm ready for the war
 +
Woah-oh-oh
 +
This is how the story goes
 +
Woah-oh-oh
 +
I guess this is how the story goes
 +
I'm in the thick of it, everybody knows
 +
They know me where it snows, I skied in and they froze
 +
I don't know no nothin' 'bout no ice, I'm just cold
 +
Forty somethin' milli' subs or so, I've been told
 +
From the screen to the ring, to the pen, to the king
 +
Where's my crown? That's my bling
 +
Always drama when I ring
 +
See, I believe that if I see it in my heart
 +
Smash through the ceiling 'cause I'm reachin' for the stars
 +
Woah-oh-oh
 +
This is how the story goes
 +
Woah-oh-oh
 +
I guess this is how the story goes
 +
I'm in the thick of it, everybody knows
 +
They know me where it snows, I skied in and they froze (woo)
 +
I don't know no nothin' 'bout no ice, I'm just cold
 +
Forty somethin' milli' subs or so, I've been told
 +
Highway to heaven, I'm just cruisin' by my lone'
 +
They cast me out, left me for dead, them people cold
 +
My faith in God, mind in the sun, I'm 'bout to sow (yeah)
 +
My life is hard, I took the wheel, I cracked the code (yeah-yeah, woah-oh-oh)
 +
Ain't nobody gon' save you, man, this life will break you (yeah, woah-oh-oh)
 +
In the thick of it, this is how the story goes
 +
I'm in the thick of it, everybody knows
 +
They know me where it snows, I skied in and they froze
 +
I don't know no nothin' 'bout no ice, I'm just cold
 +
Forty somethin' milli' subs or so, I've been told
 +
I'm in the thick of it, everybody knows (everybody knows)
 +
They know me where it snows, I skied in and they froze (yeah)
 +
I don't know no nothin' 'bout no ice, I'm just cold
 +
Forty somethin' milli' subs or so, I've been told (ooh-ooh)
 +
Woah-oh-oh (nah-nah-nah-nah, ayy, ayy)
 +
This is how the story goes (nah, nah)
 +
Woah-oh-oh
 +
I guess this is how the story goes
  
 
<math>\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }10\qquad\textbf{(E) }13</math>
 
<math>\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }10\qquad\textbf{(E) }13</math>

Revision as of 02:25, 31 January 2025

https://www.linkedin.com/in/quandalious-dingleton-380a90243/ 3.I'm in the thick of it, everybody knows They know me where it snows, I skied in and they froze I don't know no nothin' 'bout no ice, I'm just cold Forty somethin' milli' subs or so, I've been told I'm in my prime and this ain't even final form They knocked me down, but still, my feet, they find the floor I went from living rooms straight out to sold-out tours Life's a fight, but trust, I'm ready for the war Woah-oh-oh This is how the story goes Woah-oh-oh I guess this is how the story goes I'm in the thick of it, everybody knows They know me where it snows, I skied in and they froze I don't know no nothin' 'bout no ice, I'm just cold Forty somethin' milli' subs or so, I've been told From the screen to the ring, to the pen, to the king Where's my crown? That's my bling Always drama when I ring See, I believe that if I see it in my heart Smash through the ceiling 'cause I'm reachin' for the stars Woah-oh-oh This is how the story goes Woah-oh-oh I guess this is how the story goes I'm in the thick of it, everybody knows They know me where it snows, I skied in and they froze (woo) I don't know no nothin' 'bout no ice, I'm just cold Forty somethin' milli' subs or so, I've been told Highway to heaven, I'm just cruisin' by my lone' They cast me out, left me for dead, them people cold My faith in God, mind in the sun, I'm 'bout to sow (yeah) My life is hard, I took the wheel, I cracked the code (yeah-yeah, woah-oh-oh) Ain't nobody gon' save you, man, this life will break you (yeah, woah-oh-oh) In the thick of it, this is how the story goes I'm in the thick of it, everybody knows They know me where it snows, I skied in and they froze I don't know no nothin' 'bout no ice, I'm just cold Forty somethin' milli' subs or so, I've been told I'm in the thick of it, everybody knows (everybody knows) They know me where it snows, I skied in and they froze (yeah) I don't know no nothin' 'bout no ice, I'm just cold Forty somethin' milli' subs or so, I've been told (ooh-ooh) Woah-oh-oh (nah-nah-nah-nah, ayy, ayy) This is how the story goes (nah, nah) Woah-oh-oh I guess this is how the story goes

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

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 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


See also

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

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