Aptitude Discussion

Q. |
If $n = 1 + x$, where $x$ is a product of four consecutive positive integers, then which of the following is true? A. $n$ is odd B. $n$ is prime C. $n$ is a perfect square |

✔ A. |
A and C only |

✖ B. |
A and B only |

✖ C. |
A only |

✖ D. |
None of these |

**Solution:**

Option(**A**) is correct

Since $x$ is the product of four consecutive integers, it is always divisible by 4, i,.e., it is always even. So, $1 + x$ is always odd.

$n=1+x$

$\begin{align*}

x&=(y-1)(y)(y+1)(y+2)\\

&=y(y^2-1)(y+2)\\

&=(y^3-y)(y+2)\\

&=y^4+2y^3-y^2-2y

\end{align*}$

$\begin{align*}

\Rightarrow 1+x& =y^4+2y^3-y^2-2y+1 \\

&= y^4+y^2+1+2y^3-2y^2-2y\\

&=(y^2+y-1)^2

\end{align*}$

So, $1 + x$ is a perfect square as we can see. Hence, **option A** is the correct choice.

**Note: **In an exam, even if you are not able to work mathematically as above, you should not leave out this type of question. You should take two or three numerical values for $x$ and check which of the choices will be satisfiÂed. Take the four consecutive integers as $(1, 2, 3, 4)$, $(2, 3, 4, 5)$ and $(3, 4, 5, 6)$ and in each case, we find that $1+ x$ is a perfect square and odd. Then, we can mark (A) as the answer choice.