Que : If n is an even integer, which of the following MUST BE an integer?

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Que : If n is an even integer, which of the following MUST BE an integer?

I. \(\frac{\left(n^2-1\right)}{2}\) II. \(\frac{\left(n-1\right)}{2}\) III. \(\frac{n}{2}\)

A. I only
B. II only
C. III only
D. I & III only
E. I, II, & III

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Solution: Multi-Choice questions: ‘Must be’ VS ‘Could be’

We have to find the options that must be true

‘MUST BE’: Choose only the options that are absolutely true in ALL cases

=> ‘n’ is an even integer, Always an integer

=> n=even=2m (m is always an integer)

Option I: \(\frac{n^2-1}{2}=\frac{\left(\left(2m\right)^2-1\right)}{2}=\frac{\left(4m^2-1\right)}{2}=2m^2-0.5\) => NOT AN INTEGER ∴ Option is not TRUE

Option II: \(\frac{n-1}{2}=\frac{2m-1}{2}=m-0.5\) => NOT AN INTEGER ∴ Option is not TRUE

Option III: \(\frac{n}{2}=\frac{2m}{2}=m\) => INTEGER ∴ Option is always TRUE

Only option III is TRUE.

Therefore, C is the correct answer.

Answer C