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Difference between revisions of "2020 AMC 10B Problems/Problem 4"

(Created page with "==Problem== The acute angles of a right triangle are <math>a^{\circ}</math> and <math>b^{\circ}</math>, where <math>a>b</math> and both <math>a</math> and <math>b</math> are...")
 
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Since the three angles of a triangle add up to <math>180^{\circ}</math> and one of the angles is <math>90^{\circ}</math> because it's a right triangle, then <math>a^{\circ} + b^{\circ} = 90^{\circ}</math>.
 
Since the three angles of a triangle add up to <math>180^{\circ}</math> and one of the angles is <math>90^{\circ}</math> because it's a right triangle, then <math>a^{\circ} + b^{\circ} = 90^{\circ}</math>.
  
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The greatest prime number less than <math>90</math> is <math>89</math>. If <math>a=89^{\circ}</math>, then <math>b=1^{\circ}</math>, which is not prime.
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The greatest prime number less than <math>90</math> is <math>89</math>. If <math>a=89^{\circ}</math>, then <math>b=90^{\circ]-89^{\circ}=1^{\circ}</math>, which is not prime.
  
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The next greatest prime number less than <math>90</math> is <math>83</math>. If <math>a=83^{\circ}</math>, then <math>b=7^{\circ}</math>, which IS prime, so we have our answer <math>\boxed{\textbf{(D)}\ 7}</math>
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The next greatest prime number less than <math>90</math> is <math>83</math>. If <math>a=83^{\circ}</math>, then <math>b=7^{\circ}</math>, which IS prime, so we have our answer <math>\boxed{\textbf{(D)}\ 7}</math> ~quacker88

Revision as of 16:41, 7 February 2020

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 $b=90^{\circ]-89^{\circ}=1^{\circ}$ (Error compiling LaTeX. Unknown error_msg), 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

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