Difference between revisions of "2020 AMC 10B Problems/Problem 4"
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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>. | ||
− | The greatest prime number less than <math>90</math> is <math>89</math>. If <math>a=89^{\circ}</math>, then <math>b=90^{\circ | + | 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. |
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 | 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 and
, where
and both
and
are prime numbers. What is the least possible value of
?
Solution
Since the three angles of a triangle add up to and one of the angles is
because it's a right triangle, then
.
The greatest prime number less than is
. If
, then
, which is not prime.
The next greatest prime number less than is
. If
, then
, which IS prime, so we have our answer
~quacker88