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2020 AMC 10B Problem 4

Problem 4 of 25EasierGeometryNumber Theory

The acute angles of a right triangle are a∘a^{\circ} and b∘,b^{\circ}, where a>ba>b and both aa and bb are prime numbers. What is the least possible value of b?b?

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Solution

The acute angles satisfy a+b=90.a+b=90. Testing primes bb from least to greatest, b=2,3,5b=2,3,5 would give a=88,87,85,a=88,87,85, none of which is prime. For b=7,b=7, however, a=83a=83 is prime and is greater than b.b. Therefore the least possible value of bb is 7.7. Thus, D is the correct answer.

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Concepts: prime · angle sum

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.