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, $a^{\circ} + b^{\circ} = 90^{\circ}$ .

The greatest prime number less than $90$ is $89$ . If $a=89^{\circ}$ , then $b=90^{\circ}-89^{\circ}=1^{\circ}$ , 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 $\boxed{\textbf{(D)}\ 7}$ ~quacker88

Solution 2

Looking at the answer choices, only $7$ and $11$ are coprime to $90$ . Testing $7$ makes the other angle $83$ which is prime, therefore our answer is $\boxed{\textbf{(D)}\ 7}$

Video Solution

https://youtu.be/Gkm5rU5MlOU

~IceMatrix

2020 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
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All AMC 10 Problems and Solutions

2020 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
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All AMC 12 Problems and Solutions

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2020 AMC 10B Problems/Problem 4

Contents

Problem

Solution

Solution 2

Video Solution

See Also