GMAT Practice Question: If k is a positive even integer, which of the following must be an odd integer? I....
Question
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
- II only
- III only
- I and III only
- II and III only
- I, II, and III
Topics: odds/evens, exponents
Solution
Step 1
We start by expressing k as twice an integer to reflect that it is even.
k = 2n \text{ for some integer } n
Step 2
We substitute k = 2n into k^{2} - 3k + 4 and factor out 2 to test its parity.
(2n)^{2} - 3(2n) + 4 = 4n^{2} - 6n + 4 =2(2n^{2} - 3n + 2) =2\times(\text{integer}) 4n^{2} - 6n + 4 \text{ is even}
Step 3
We substitute k = 2n into k^{5} + 3 and evaluate its parity.
(2n)^{5} + 3 = 32n^{5} + 3 32n^{5} \text{ is even} 32n^{5} + 3 \text{ is odd}
Step 4
We substitute k = 2n into 7k - 7 and evaluate its parity.
7(2n) - 7 = 14n - 7 14n \text{ is even} 7 \text{ is odd} 14n - 7 \text{ is odd}
Step 5
We see that only the second and third expressions must be odd, so the correct answer is choice D.
\text{Answer: D}
Answer
D
