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Difference between revisions of "2014 AMC 8 Problems/Problem 13"

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==Solution==
 
==Solution==
 
Since <math>n^2+m^2</math> is even, either both <math>n^2</math> and <math>m^2</math> are even, or they are both odd. Therefore, <math>n</math> and <math>m</math> are either both even or both odd, since the square of an even number is even and the square of an odd number is odd. As a result, <math>n+m</math> must be even. The answer, then, is  <math>n^2+m^2</math> <math>\boxed{(\text{D})n+m is odd}</math>.
 
Since <math>n^2+m^2</math> is even, either both <math>n^2</math> and <math>m^2</math> are even, or they are both odd. Therefore, <math>n</math> and <math>m</math> are either both even or both odd, since the square of an even number is even and the square of an odd number is odd. As a result, <math>n+m</math> must be even. The answer, then, is  <math>n^2+m^2</math> <math>\boxed{(\text{D})n+m is odd}</math>.
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==Solution 2==
 
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One of the solutions above (Not by me. It is named Solution) is incorrect, because the answer is <math>\boxed{D}</math>. If we replace <math>n</math> and <math>m</math> with odd numbers, the value outputs to a even same with inputting with even numbers. This crosses out A and B. It also crosses out C because odd + odd is even. For D, we have to have one odd and one even. If we take 1 and 2 for an example - <math>1^2</math>+<math>2^2</math>= 1+4= 5 which is odd. We can also see that odd * odd is odd, and even * even is even, and odd + even is odd. This crosses out E, and we get <math>\boxed{D}</math>.
 
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~[[User:Multpi12|Multpi12]]
 
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2014|num-b=12|num-a=14}}
 
{{AMC8 box|year=2014|num-b=12|num-a=14}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 22:20, 7 May 2024

Problem

If $n$ and $m$ are integers and $n^2+m^2$ is even, which of the following is impossible?

$\textbf{(A) }$ $n$ and $m$ are even $\qquad\textbf{(B) }$ $n$ and $m$ are odd $\qquad\textbf{(C) }$ $n+m$ is even $\qquad\textbf{(D) }$ $n+m$ is odd $\qquad \textbf{(E) }$ none of these are impossible


Video Solution (CREATIVE THINKING)

https://youtu.be/jQLIxT5qCTY

~Education, the Study of Everything


Video Solution

https://www.youtube.com/watch?v=boXUIcEcAno ~David

https://youtu.be/_3n4f0v6B7I ~savannahsolver

Solution

Since $n^2+m^2$ is even, either both $n^2$ and $m^2$ are even, or they are both odd. Therefore, $n$ and $m$ are either both even or both odd, since the square of an even number is even and the square of an odd number is odd. As a result, $n+m$ must be even. The answer, then, is $n^2+m^2$ $\boxed{(\text{D})n+m is odd}$.

See Also

2014 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
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 AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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