Difference between revisions of "2008 AMC 10B Problems/Problem 15"
Pi is 3.14 (talk | contribs) (→Solution) |
Aopsthedude (talk | contribs) (→Solution) |
||
(15 intermediate revisions by 3 users not shown) | |||
Line 13: | Line 13: | ||
We also know that <math>a^2</math> is odd and thus <math>a</math> is odd, since the right side of the equation is odd. <math>2b</math> is even. <math>2b+1</math> is odd. | We also know that <math>a^2</math> is odd and thus <math>a</math> is odd, since the right side of the equation is odd. <math>2b</math> is even. <math>2b+1</math> is odd. | ||
− | So <math>a=3,5,7,9,11,13</math>, | + | So <math>a=1,3,5,7,9,11,13</math>, but if <math>a=1</math>, then <math>b=0</math>. Thus <math>a\neq1.</math> |
+ | |||
+ | <math>a=3,5,7,9,11,13</math> | ||
+ | |||
+ | The answer is <math>\boxed{A}</math>. | ||
− | ~qkddud | + | ~qkddud (edited by aopsthedude and bburubburu) |
== Video Solution by OmegaLearn == | == Video Solution by OmegaLearn == |
Latest revision as of 10:39, 8 October 2023
Problem
How many right triangles have integer leg lengths and and a hypotenuse of length , where ?
Solution
By the Pythagorean theorem,
This means that .
We know that and that .
We also know that is odd and thus is odd, since the right side of the equation is odd. is even. is odd.
So , but if , then . Thus
The answer is .
~qkddud (edited by aopsthedude and bburubburu)
Video Solution by OmegaLearn
https://youtu.be/euz1azVKUYs?t=135
~ pi_is_3.14
See also
2008 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.