Question

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

$2$

$3$

$5$

$7$

$11$

Solution:

We know that the interior angles of a triangle add up to $180^{\circ},$ and since the triangle in question is a right triangle, by definition one of the interior angles must measure $90^{\circ}.$ The remaining two acute angles, $a^{\circ}$ and $b^{\circ},$ must therefore have a sum of $180^{\circ}-90^{\circ}=90^{\circ}.$

Let's begin by exploring the largest values of $a^{\circ}$ and going from there, as those will naturally yield the smallest values of $b^{\circ}.$

The greatest possible value of $a^{\circ}$ is $89^{\circ}.$ This makes $b^{\circ}=90^{\circ}-89^{\circ}=1^{\circ},$ which is not prime, so this is not a valid possible case.

Moving on, the next largest possible value of $a^{\circ}$ is $83^{\circ}.$ This makes $b^{\circ}=90^{\circ}-83^{\circ}=7^{\circ},$ which is prime! Therefore, this is the smallest possible value of $b.$

Thus, D is the correct answer.

2020 AMC 10B Problem 4

Solution:

Problem 4 in Other Years