2014 AMC 8 Problems/Problem 13
Problem
If and
are integers and
is even, which of the following is impossible?
and
are even
and
are odd
is even
is odd
none of these are impossible
Video Solution
https://www.youtube.com/watch?v=boXUIcEcAno
https://youtu.be/_3n4f0v6B7I ~savannahsolver
Solution
Since is even, either both
and
are even, or they are both odd. Therefore,
and
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,
must be even. The answer, then, is
.
Solution 2
Instead of using logic to solve this, we can just plug in random numbers. Since which is even, we see that it is possible for both
and
to be even, and for
to be even.
.
~Trex226 .
- note that this solution (which gave a wrong answer) involves plugging in random numbers, which is difficult for this particular problem due to answer choice E
~awesomeguy856
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 |
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