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

This is a simple question on the properties of positive integers. However, be careful about the fact that it’s a ‘Must be’ question, so the option that you pick should be true in all cases (i.e. it should be ODD in all cases)

Here are some of the concepts that will be used in solving this question:

The general form of an even number is 2k, where k is an integer. The general form of an odd number is 2k + 1 or 2k – 1, where k is an integer.

Odd * Odd = Odd
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 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, \(\frac{n}{3}\) CAN be even. For example, if n = 6, \(\frac{n}{3}\) = 2. So, \(\frac{n}{3}\) is not odd in all cases. Option D can be eliminated.

Depending on the value of n, \(\frac{n}{2}\) CAN be even. For example, if n = 4, \(\frac{n}{2}\) = 2. So, \(\frac{n}{2}\) is not odd in all cases. Option E can be eliminated.

Answer option B is the right answer.

An alternative approach would be to take a couple of simple values for n and eliminate options which are not true always. The option that gets left out HAS to be the answer.

In questions like these, it’s important to go slow when checking the options. This is because, this is a sub 600 level question and so, a lot of you will try such questions when you are in the initial stages of your preparation, due to which your concepts might be a little shaky. So, do not be in a hurry to get to the answer.

Hope this helps!