Question

Assertion :$n^2+n$ is divisible by $2$ for every positive integer $n$. Reason: If $x$ and $y$ are odd positive integers, then $x^2+y^2$ is divisible by $4$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
• B. Both Assertion and Reason are correct, but Reason is not the correct explanation for Assertion.
• C. Assertion is correct, but Reason is incorrect.
• D. Assertion is incorrect, but Reason is correct.

Answer 1

The correct option is C Assertion is correct, but Reason is incorrect.

$n^2+n=n(n+1)$
Thus if n is odd then $n+1$ will be even
If n is even then $n(n+1)$ will be odd.
As product of an even and odd number is always even, thus in both the above cases, product is divisible by 2
$x^2+y^2$ where x and y are odd positive integers
$x^2+y^2=(x+y{)}^2-2xy$
$(odd+odd{)}^2-2(odd\ast odd)$
We can see that the number is divisible by 2 but not by 4.
Option c