Question

If the product of four consecutive natural numbers increased by a natural number $\rho$ is a perfect square, then the value of $\rho$ is:

• A. 8
• B. 4
• C. 2
• D. 1

Answer 1

The correct option is D 1

Let the four consecutive natural numbers be $x,(x+1),(x+2),(x+3)$.
Now, $x(x+1)(x+2)(x+3)+p=x(x+3)(x+1)(x+2)+p$
$=(x^2+3x)(x^2+3x+2)+p$
$=(x^2+3x{)}^2+2(x^2+3x)+p$
Given that this is a perfect square, the value of $p=1$, so that the above
equation becomes $(x^2+3x{)}^2+2(x^2+3x)+1=(x^2+3x+1{)}^2$