Difference between revisions of "2024 AMC 10A Problems/Problem 3"
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<math>\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }13</math> | <math>\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }13</math> | ||
| − | == Solution == | + | == 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> | 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 | ~MRENTHUSIASM | ||
| + | |||
| + | == Solution 2 == | ||
| + | We notice that the minimum possible value of the sum of <math>5</math> 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 | ||
| + | |||
==See also== | ==See also== | ||
{{AMC10 box|year=2024|ab=A|num-b=2|num-a=4}} | {{AMC10 box|year=2024|ab=A|num-b=2|num-a=4}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 17:04, 8 November 2024
Contents
Problem
What is the sum of the digits of the smallest prime that can be written as a sum of 
distinct primes?
Solution 1
Let the requested sum be 
Recall that 
is the only even (and the smallest) prime, so 
is odd. It follows that the five distinct primes are all odd. The first few odd primes are 
We conclude that 
as 
is a composite. The next possible value of is 
which is a prime. Therefore, we have 
and the sum of its digits is 
~MRENTHUSIASM
Solution 2
We notice that the minimum possible value of the sum of distinct primes is 
, which is not a prime. The smallest prime greater than that is 
. However, this cannot be written as the sum of distinct primes, since 
is not prime. However, 
can be written as 
, so the answer is 
~andliu766
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 | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
