2014 AMC 8 Problems/Problem 13

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

Solution 2

One of the solutions above (Not by me. It is named Solution) is incorrect, because the answer is $\boxed{D}$. If we replace $n$ and $m$ 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 - $1^2$+$2^2$= 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 $\boxed{D}$.


~Multpi12

See Also

2014 AMC 8 (ProblemsAnswer KeyResources)
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

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