2002 AMC 12A Problems/Problem 17: Difference between revisions
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== Problem == | == Problem == | ||
Several sets of prime numbers, such as <math>\{7,83,421,659\}</math> use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have? | <!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>Several sets of prime numbers, such as <math>\{7,83,421,659\}</math> use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude> | ||
<math> | <math> | ||
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Neither of the digits <math>4</math>, <math>6</math>, and <math>8</math> can be a units digit of a prime. Therefore the sum of the set is at least <math>40 + 60 + 80 + 1 + 2 + 3 + 5 + 7 + 9 = 207</math>. | Neither of the digits <math>4</math>, <math>6</math>, and <math>8</math> can be a units digit of a prime. Therefore the sum of the set is at least <math>40 + 60 + 80 + 1 + 2 + 3 + 5 + 7 + 9 = 207</math>. | ||
We can indeed create a set of primes with this sum, for example the following | We can indeed create a set of primes with this sum, for example the following sets work: <math>\{ 41, 67, 89, 2, 3, 5 \}</math> or <math>\{ 43, 61, 89, 2, 5, 7 \}</math>. | ||
Thus the answer is <math>\boxed{(B) | Thus the answer is <math>207\implies \boxed{\mathrm{(B)}}</math>. | ||
== Video Solution == | |||
https://www.youtube.com/watch?v=V6z7GiitBUM | |||
== See Also == | == See Also == | ||
{{AMC12 box|year=2002|ab=A|num-b=16|num-a=18}} | {{AMC12 box|year=2002|ab=A|num-b=16|num-a=18}} | ||
{{MAA Notice}} | |||
Latest revision as of 14:01, 8 September 2022
Problem
Several sets of prime numbers, such as 
use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?
Solution
Neither of the digits 
, 
, and 
can be a units digit of a prime. Therefore the sum of the set is at least 
.
We can indeed create a set of primes with this sum, for example the following sets work: 
or 
.
Thus the answer is 
.
Video Solution
https://www.youtube.com/watch?v=V6z7GiitBUM
See Also
| 2002 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 16 |
Followed by Problem 18 |
| 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 12 Problems and Solutions | |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing
