2020 AMC 10B Problems/Problem 4

Problem

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 $b$ ?

$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 11$

Solution

Since the three angles of a triangle add up to $180^{\circ}$ and one of the angles is $90^{\circ}$ because it's a right triangle, then $a^{\circ} + b^{\circ} = 90^{\circ}$ .

The greatest prime number less than $90$ is $89$ . If $a=89^{\circ}$ , then $b=1^{\circ}$ , which is not prime.

The next greatest prime number less than $90$ is $83$ . If $a=83^{\circ}$ , then $b=7^{\circ}$ , which IS prime, so we have our answer $\boxed{\textbf{(D)}\ 7}$

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2020 AMC 10B Problems/Problem 4

Problem

Solution