Difference between revisions of "2005 AMC 12A Problems/Problem 18"
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== Problem == | == Problem == | ||
− | + | Call a number ''prime-looking'' if it is [[composite]] but not divisible by <math>2, 3,</math> or <math>5.</math> The three smallest prime-looking numbers are <math>49, 77</math>, and <math>91</math>. There are <math>168</math> prime numbers less than <math>1000</math>. How many prime-looking numbers are there less than <math>1000</math>? | |
<math> | <math> | ||
− | (\mathrm {A}) \ | + | (\mathrm {A}) \ 100 \qquad (\mathrm {B}) \ 102 \qquad (\mathrm {C})\ 104 \qquad (\mathrm {D}) \ 106 \qquad (\mathrm {E})\ 108 |
</math> | </math> | ||
== Solution == | == Solution == | ||
− | <math> | + | The given states that there are <math>168</math> prime numbers less than <math>1000</math>, which is a fact we must somehow utilize. Since there seems to be no easy way to directly calculate the number of "prime-looking" numbers, we can apply [[complementary counting]]. We can split the numbers from <math>1</math> to <math>1000</math> into several groups: <math>\{1\},</math> <math>\{\mathrm{numbers\ divisible\ by\ 2 = S_2}\},</math> <math> \{\mathrm{numbers\ divisible\ by\ 3 = S_3}\},</math> <math> \{\mathrm{numbers\ divisible\ by\ 5 = S_5}\}, \{\mathrm{primes\ not\ including\ 2,3,5}\},</math> <math> \{\mathrm{prime-looking}\}</math>. Hence, the number of prime-looking numbers is <math>1000 - (168-3) - 1 - |S_2 \cup S_3 \cup S_5|</math> (note that <math>2,3,5</math> are primes). |
+ | |||
+ | We can calculate <math>S_2 \cup S_3 \cup S_5</math> using the [[Principle of Inclusion-Exclusion]]: (the values of <math>|S_2| \ldots</math> and their intersections can be found quite easily) | ||
+ | |||
+ | <div style="text-align:center;"><math>|S_2 \cup S_3 \cup S_5| = |S_2| + |S_3| + |S_5| - |S_2 \cap S_3| - |S_3 \cap S_5| - |S_2 \cap S_5| + |S_2 \cap S_3 \cap S_5|</math><br /><math>= 500 + 333 + 200 - 166 - 66 - 100 + 33 = 734</math></div> | ||
+ | |||
+ | Substituting, we find that our answer is <math>1000 - 165 - 1 - 734 = 100 \Longrightarrow \mathrm{(A)}</math>. | ||
== See also == | == See also == | ||
− | {{AMC12 box|year=2005| | + | {{AMC12 box|year=2005|num-b=17|num-a=19|ab=A}} |
− | [[Category:Introductory | + | [[Category:Introductory Number Theory Problems]] |
+ | {{MAA Notice}} |
Latest revision as of 17:15, 25 November 2020
Problem
Call a number prime-looking if it is composite but not divisible by or The three smallest prime-looking numbers are , and . There are prime numbers less than . How many prime-looking numbers are there less than ?
Solution
The given states that there are prime numbers less than , which is a fact we must somehow utilize. Since there seems to be no easy way to directly calculate the number of "prime-looking" numbers, we can apply complementary counting. We can split the numbers from to into several groups: . Hence, the number of prime-looking numbers is (note that are primes).
We can calculate using the Principle of Inclusion-Exclusion: (the values of and their intersections can be found quite easily)
Substituting, we find that our answer is .
See also
2005 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
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 |
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