Problem:
The acute angles of a right triangle are and , where and both and are prime numbers. What is the least possible value of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Since the three angles of a triangle add up to and one of the angles is because it's a right triangle, a^{\circ}+b^{\circ}=90^
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 .
OR
Looking at the answer choices, only and are coprime to . Testing , the smaller angle, makes the other angle 83 which is prime, therefore our answer is .
OR
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 .
The problems on this page are the property of the MAA's American Mathematics Competitions