2020 AMC 10B Problems/Problem 4
- The following problem is from both the 2020 AMC 10B #4 and 2020 AMC 12B #4, so both problems redirect to this page.
Contents
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 1
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
Solution 3 (Euclidean Algorithm)
It is clear that By the Euclidean Algorithm, we have so and are relatively prime.
The least such prime number is from which is also a prime number. Therefore, the answer is
~MRENTHUSIASM
Video Solutions
~IceMatrix
~savannahsolver
~AlexExplains
~Education, the Study of Everything
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.