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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

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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 satisfied. 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.


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