If $k$ is a positive even integer, which of the fo... | Free GMAT Practice Question | GMAT Panda

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If k is a positive even integer, which of the following must be an odd integer? I. k^{2} - 3 k + 4 II. k^{5} + 3 III. 7 k - 7

Explanation	Calculations	Help
$We start by expressing k as twice an integer to reflect that it is even.$	$k = 2 n for some integer n$
$We substitute k = 2 n into k^{2} - 3 k + 4 and factor out 2 to test its parity.$	$(2 n)^{2} - 3 (2 n) + 4 = 4 n^{2} - 6 n + 4 = 2 (2 n^{2} - 3 n + 2) = 2 \times (integer) 4 n^{2} - 6 n + 4 is even$
$We substitute k = 2 n into k^{5} + 3 and evaluate its parity.$	$(2 n)^{5} + 3 = 32 n^{5} + 3 32 n^{5} is even 32 n^{5} + 3 is odd$	Theory & Tactics Method Card OE1-C Click to view full details
$We substitute k = 2 n into 7 k - 7 and evaluate its parity.$	$7 (2 n) - 7 = 14 n - 7 14 n is even 7 is odd 14 n - 7 is odd$	Theory & Tactics Method Card OE3-D Click to view full details
$We see that only the second and third expressions must be odd, so the correct answer is choice D.$	$Answer: D$

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D

If k is a positive even integer, which of the following must be an odd integer? I. k^{2} - 3 k + 4 II. k^{5} + 3 III. 7 k - 7

Explanation	Calculations	Help
$We start by expressing k as twice an integer to reflect that it is even.$	$k = 2 n for some integer n$
$We substitute k = 2 n into k^{2} - 3 k + 4 and factor out 2 to test its parity.$	$(2 n)^{2} - 3 (2 n) + 4 = 4 n^{2} - 6 n + 4 = 2 (2 n^{2} - 3 n + 2) = 2 \times (integer) 4 n^{2} - 6 n + 4 is even$
$We substitute k = 2 n into k^{5} + 3 and evaluate its parity.$	$(2 n)^{5} + 3 = 32 n^{5} + 3 32 n^{5} is even 32 n^{5} + 3 is odd$	Theory & Tactics Method Card OE1-C Click to view full details
$We substitute k = 2 n into 7 k - 7 and evaluate its parity.$	$7 (2 n) - 7 = 14 n - 7 14 n is even 7 is odd 14 n - 7 is odd$	Theory & Tactics Method Card OE3-D Click to view full details
$We see that only the second and third expressions must be odd, so the correct answer is choice D.$	$Answer: D$

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D