2014 AMC 8 Problem 13
Attempt Problem 13 of the 2014 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2014 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
13.
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
Answer: D
Small hint:
A square has the same parity as its base integer.
Big hint:
For a sum of two squares to be even, the two integers have the same parity.
Solution:
Since is even, either and are both odd, or both even.
If they are both odd, then and are both odd. If they are both even, then and are both even. If and are both odd or even, their sum will always be even.
Therefore, is never odd.
Thus, D is the correct answer.
Problem 13 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
- 1988 AMC 8
- 1989 AMC 8
- 1990 AMC 8
- 1991 AMC 8
- 1992 AMC 8
- 1993 AMC 8
- 1994 AMC 8
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- 1998 AMC 8
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