Question

For an integer n, a student states the following:
I. If $n$ is odd, $(n+1{)}^2$ is even
II. If $n$ is even, $(n-1{)}^2$ is odd
III. If $n$ is even, $\sqrt{n-1}$ is irrational.
Which of the above statements would be true?

• A. I and III
• B. I and II
• C. I, II and III
• D. II and III

Answer 1

The correct option is B

I and II

$\sqrt{x}$ is defined for a number $x$ only if $x$ is positive.
I - If $n$ is odd, then $n+1$ is even, which implies that ${(n+1)}^2$ is even.
II - If $n$ is even, then $n-1$ is odd, which implies that ${(n-1)}^2$ is odd.
III - If (n - 1) < 0 then $\sqrt{n-1}$ is not defined. Thus, this statement is false.

Hence, I and II are the two corrrect statements.