## 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?

by [email protected] Revolution » Wed Apr 07, 2021 9:29 pm

00:00

A

B

C

D

E

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

by [email protected] Revolution » Tue Apr 13, 2021 12:18 am
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.