Bunuel wrote:
If a is an odd integer, which of the following must be an even integer?
A. a^4−a+1
B. (a^4−a)(a+1/a)
C. a^4−a^3+a^2+2a
D. (a^3+a^2+a)^2
E. None of the above.
If we add or subtract three odd numbers, we get an odd number. So A is odd, and answer C becomes odd + 2a, or odd + even, which is odd, and answer D becomes odd^2, which is odd. So the answer is B or E.
B looks implausible at first glance, because it includes a fraction, so it looks as though B might not always be an integer at all. But if we rewrite B by factoring out one "a" from the first factor, then multiplying that "a" through the second factor, we can see that the denominator will always disappear:
\(
(a^4 - a)(a + \frac{1}{a} ) = (a^3 - 1)(a)(a + \frac{1}{a}) = (a^3 - 1)(a^2 + 1)
\)
and now we're multiplying the two integers a^3 - 1, which is even (odd - odd is even), and a^2 + 1, which is also even (odd + odd is even), so the product will be even.
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