The acute angles of a right triangle are a° and b°, where a \gt b and both a and b are prime numbers. What is the least possible value of b?

2020 AMC 12B Problem 4 — Solution

2020 AMC 12B Problem 4

Below is the professionally curated solution for Problem 4 of the 2020 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2020 AMC 12B solutions, or check the answer key.

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Concepts:prime angle sum

Difficulty rating: 1220

The acute angles of a right triangle are $a^\circ$ and $b^\circ,$ where $a \gt b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b?$

$2$

$3$

$5$

$7$

$11$

Solution:

Since the angles are complementary, $a + b = 90.$ To minimize $b,$ try small primes and require $90 - b$ to be prime as well.

For $b = 2, 3, 5,$ the value $90 - b = 88, 87, 85$ is not prime. For $b = 7,$ we get $90 - 7 = 83,$ which is prime. So the least possible value is $b = 7.$

Thus, the correct answer is D.

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