Question: If n is an even integer, which of the following must be an odd integer?

1. 3n - 2
2. 3 (n + 1)
3. n - 2
4. \(n\over3\)
5. \(n\over2\)

Answer:
Approach Solution (1):
The general form of an even number is 2k, where k is an integer. The general form of an odd numbers is 2k + 1 or 2k – 1, where k is an integer
Odd * Odd = Even
Odd * Even = Even
Even * Even = Even
Odd + Odd = Even
Even + Even = Even
Odd + Even = Odd (the same equations hold good even when the addition symbols are replaced by subtraction symbols)
Since n is an even integer, let n = 2k. Let us plug this value in the options and check which options can be eliminated
3n – 2 = 6k – 2 = Even – Even = Even
So the expression given in the option A will always be even
Option A can be eliminated since the expression will never be an odd number
3 (n + 1) = 3 (2k + 1) = Odd * Odd = Odd
This means, the expression given in option B will always be odd, regardless of the value of k. let’s hold on to option B
n – 2 = 2k – 2 = Even – Even = Even
Similar to option A, option C is also always even. Option C can be eliminated
Depending on the value of n, \(n\over3\)can be even
Option D is eliminated
Depending on the value of n, \(n\over2\)can be even
Option E can be eliminated
Correct option: B

Approach Solution (2):
Consider even integer = 2
Option A = (3 × 2 − 2) = 4 = Even integer
Option C = 2 – 2 = 0 = Even integer
Option B = 3(2 + 1) = 9 = Odd integer
Option D = 2 × 2 = 4 = Even integer
Correct answer is option B
Correct option: B

Approach Solution (3):
Since n is even, 3(n + 1) must be odd since 3 is odd and (n + 1) is also odd.
Odd * Odd = odd
Correct option: B

“If n is an even integer, which of the following must be an odd integer?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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