Question

The product of two consecutive positive integers is divisible by $2$. Using this theory, consider $p$ and $(p+1)$ as two consecutive numbers where $p=2q+1$ and $q=4.$ Then which of the following is a factor of the product.

• A. $54$
• B. $56$
• C. $45$
• D. $55$

Answer 1

The correct option is C $45$

Let the two consecutive numbers be $p$ and $(p+1)$

$p+1$ is odd number .... (given)
$\therefore p=2q+1$
Now, $p(p+1)=p^2+p=(2q+1{)}^2+(2q+1)$
$=4q^2+4q+1+2q+1$
$=4q^2+6q+2$
$=\left(4\right(4^2)+6(4)+2)=4\left(16\right)+24+2=64+24+2=90$

$\therefore 90÷2=45$ ....since product is divisible by $2$

So, option C is correct.