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

[Max@Math Revolution]

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

I. \(\frac{n^2-1}{2}\)

II. \(\frac{n-1}{2}\)

III. \(\frac{n}{2}\)


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

[Max@Math Revolution]

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 odd integer, Always an integer

=> n = odd = 2m+1 (m is always an integer)

I. \(\frac{n^2-1}{2}=\frac{\left[\left(2m+1\right)^2-1\right]}{2}=\frac{\left[4m^2+4m+1-1\right]}{2}=2m^2+m\) => AN INTEGER ∴ Option is TRUE

II. \(\frac{n-1}{2}=\frac{\left[\left(2m+1\right)-1\right]}{2}=m\) => AN INTEGER ∴ Option is TRUE

III. \(\frac{n}{2}=\frac{\left(2m+1\right)}{2}=m+0.5\) => NOT AN INTEGER ∴ Option is NOT TRUE

Only options I and II are TRUE.


Therefore, D is the correct answer.

Answer D