Difference between revisions of "2020 AMC 10B Problems/Problem 4"
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+ | {{duplicate|[[2020 AMC 10B Problems|2020 AMC 10B #4]] and [[2020 AMC 12B Problems|2020 AMC 12B #4]]}} | ||
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==Problem== | ==Problem== | ||
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==Solution== | ==Solution== | ||
− | Since the three angles of a triangle add up to <math>180^{\circ}</math> and one of the angles is <math>90^{\circ}</math> because it's a right triangle, | + | Since the three angles of a triangle add up to <math>180^{\circ}</math> and one of the angles is <math>90^{\circ}</math> because it's a right triangle, <math>a^{\circ} + b^{\circ} = 90^{\circ}</math>. |
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+ | The greatest prime number less than <math>90</math> is <math>89</math>. If <math>a=89^{\circ}</math>, then <math>b=90^{\circ}-89^{\circ}=1^{\circ}</math>, which is not prime. | ||
+ | |||
+ | The next greatest prime number less than <math>90</math> is <math>83</math>. If <math>a=83^{\circ}</math>, then <math>b=7^{\circ}</math>, which IS prime, so we have our answer <math>\boxed{\textbf{(D)}\ 7}</math> ~quacker88 | ||
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+ | ==Solution 2== | ||
+ | |||
+ | Looking at the answer choices, only <math>7</math> and <math>11</math> are coprime to <math>90</math>. Testing <math>7</math>, the smaller angle, makes the other angle <math>83</math> which is prime, therefore our answer is <math>\boxed{\textbf{(D)}\ 7}</math> | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/Gkm5rU5MlOU | ||
+ | |||
+ | ~IceMatrix | ||
+ | |||
+ | https://youtu.be/y_nsQ7pO63c | ||
+ | |||
+ | ~savannahsolver | ||
− | + | ==See Also== | |
− | + | {{AMC10 box|year=2020|ab=B|num-b=3|num-a=5}} | |
+ | {{AMC12 box|year=2020|ab=B|num-b=3|num-a=5}} | ||
+ | {{MAA Notice}} |
Revision as of 12:47, 17 June 2020
- The following problem is from both the 2020 AMC 10B #4 and 2020 AMC 12B #4, so both problems redirect to this page.
Problem
The acute angles of a right triangle are and , where and both and are prime numbers. What is the least possible value of ?
Solution
Since the three angles of a triangle add up to and one of the angles is because it's a right triangle, .
The greatest prime number less than is . If , then , which is not prime.
The next greatest prime number less than is . If , then , which IS prime, so we have our answer ~quacker88
Solution 2
Looking at the answer choices, only and are coprime to . Testing , the smaller angle, makes the other angle which is prime, therefore our answer is
Video Solution
~IceMatrix
~savannahsolver
See Also
2020 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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 |
2020 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 3 |
Followed by Problem 5 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.