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

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==Problem==
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==Problem 13==
 
If <math>n</math> and <math>m</math> are integers and <math>n^2+m^2</math> is even, which of the following is impossible?
 
If <math>n</math> and <math>m</math> are integers and <math>n^2+m^2</math> is even, which of the following is impossible?
  
 
<math>\textbf{(A) }</math> <math>n</math> and <math>m</math> are even <math>\qquad\textbf{(B) }</math> <math>n</math> and <math>m</math> are odd <math>\qquad\textbf{(C) }</math> <math>n+m</math> is even <math>\qquad\textbf{(D) }</math> <math>n+m</math> is odd <math>\qquad \textbf{(E) }</math> none of these are impossible
 
<math>\textbf{(A) }</math> <math>n</math> and <math>m</math> are even <math>\qquad\textbf{(B) }</math> <math>n</math> and <math>m</math> are odd <math>\qquad\textbf{(C) }</math> <math>n+m</math> is even <math>\qquad\textbf{(D) }</math> <math>n+m</math> is odd <math>\qquad \textbf{(E) }</math> none of these are impossible
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==Video Solution (CREATIVE THINKING)==
 
==Video Solution (CREATIVE THINKING)==
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==Solution==
 
==Solution==
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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{C}</math>.(Typo in the answer)
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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})}</math> is odd.
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==Solution 2==
 
==Solution 2==
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This solution above (not by me) 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 - 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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- Multpi12
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We can assign values of n and m as 1,3. Then 1^2<math> + 3^2</math> is 10. We can also assign values of n and m to be even, like 2 and 4. After calculating, we can determine that <math>n^2</math> + <math>m^2</math> could be odd or even. Then, we can conclude that n+m is not odd. Hence our answer choice is <math>\boxed{(\text{D})}</math>-TheNerdWhoIsNerdy.
  
 
==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}}

Latest revision as of 12:22, 15 September 2024

Problem 13

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 $n^2+m^2$ $\boxed{(\text{D})}$ is odd.


Solution 2

We can assign values of n and m as 1,3. Then 1^2$+ 3^2$ is 10. We can also assign values of n and m to be even, like 2 and 4. After calculating, we can determine that $n^2$ + $m^2$ could be odd or even. Then, we can conclude that n+m is not odd. Hence our answer choice is $\boxed{(\text{D})}$-TheNerdWhoIsNerdy.

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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