Question

State true or false:

'$n^2+n$' is divisible by $2$ for any positive integer '$n$'.

• A. True
• B. False

Answer 1

Answer: (A)

Since, By Euclid's Division lemma, "Any positive integer of form $2q$ or $2q+1$ where $q$ is some integer

Case 1: $n=2q$

$n^2+n$

$={\left(2q\right)}^2+2q$

$=4q^2+2q$

$=2q(2q+1)$

which is divisible by $2$.

Case 2: $n=2q+1$

$n^2+n$

$={(2q+1)}^2+2q+1$

$=4q^2+1+4q+2q+1$

$=4q^2+6q+2$

$=2(2q^2+3q+1)$

which is divisible by $2$.

From case 1, case 2

$n^2+n$ is divisible by $2$ for every positive integer $n$.

Hence, the given statement is True.