AbdurRakib
Is the positive integer n odd?
(1) \(n^2+(n+1)^2+(n+2)^2\) is even
(2) \(n^2-(n+1)^2-(n+2)^2\) is even
OG Q 2017 New Question(Book Question: 194)
Solution:
Question Stem Analysis:We need to determine whether positive integer n is odd.
Let’s review a few even/odd rules:
If n is odd, then (n + 1) is even, and (n + 2) is odd
Odd x odd = odd, even x even = even, and even x odd = even
Odd + odd = even, even + even = even, and odd + even = odd
Statement One Alone:If n is odd, then n^2 is odd, (n + 1)^2 is even, and (n + 2)^2 is odd. The sum n^2 + (n + 1)^2 + (n + 2)^2 = odd + even + odd = even.
If n is even, then n^2 is even, (n + 1)^2 is odd, and (n + 2)^2 is even. The sum n^2 + (n + 1)^2 + (n + 2)^2 = even + odd + even = odd.
From the above, we see that, in order for n^2 + (n + 1)^2 + (n + 2)^2 to be even,then n must be odd. Statement one alone is sufficient.
Statement Two Alone:If n is odd, then n^2 is odd, (n + 1)^2 is even, and (n + 2)^2 is odd and n^2 - (n + 1)^2 - (n + 2)^2 = odd - even - odd = even.
If n is even, then n^2 is even, (n + 1)^2 is odd, and (n + 2)^2 is even and n^2 - (n + 1)^2 - (n + 2)^2 = even - odd - even = odd.
From the above, we see that in order for n^2 - (n + 1)^2 - (n + 2)^2 to be even, then n must be odd. Statement two alone is sufficient.
Answer: D