2024 AMC 10A Problems/Problem 3

Problem

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

$\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }13$

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 $5$ 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

2024 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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2024 AMC 10A Problems/Problem 3

Contents

Problem

Solution 1

Solution 2

See also