Question

Given \[n = 1 + x\] and \[x\] is the product of four consecutive integers. Then n will be
(More than one option may be correct)
A.n is an odd number
B.n is prime
C.n is sometimes a perfect square
D.All of the above

Answer 1

Hint:In order to solve the problem first we will consider the consecutive numbers given in the problem statement in terms of some general variables. Further we will use the same in order to satisfy the given options and find the solution of the problem.

Complete step-by-step answer:
As we know that the difference between the consecutive numbers is 1.
Let the first consecutive number is \[p\]
Further the other three consecutive numbers will be \[p + 1,p + 2,p + 3\]
So the consecutive numbers are
\[p,p + 1,p + 2,p + 3\]
Let us find the product of these four consecutive numbers so we get
\[
  p \times \left( {p + 1} \right) \times \left( {p + 2} \right) \times \left( {p + 3} \right) \\
   = \left( {{p^2} + p} \right) \times \left( {{p^2} + 5p + 6} \right) \\
   = {p^4} + 6{p^3} + 11{p^2} + 6p \\
 \]
So we have the term x is:
\[x = {p^4} + 6{p^3} + 11{p^2} + 6p\]
From the above product as we know that the product of four consecutive numbers is even.
So further we have the required number n as:
\[
  \because x = {p^4} + 6{p^3} + 11{p^2} + 6p \\
  \because n = x + 1 \\
   \Rightarrow n = {p^4} + 6{p^3} + 11{p^2} + 6p + 1 \\
 \]
So, the term n will be odd as it is one more than the even number x.
Let us check if the number n is a perfect square or not.
Let us try to complete the square of given number n by basic completing square method.
\[
  \because n = {p^4} + 6{p^3} + 11{p^2} + 6p + 1 \\
   \Rightarrow n = {\left( {{p^2}} \right)^4} + \left( {6{p^2} \times p} \right) + \left( {11{p^2} \times 1} \right) + \left( {6p \times 1} \right) + {1^2} \\
   \Rightarrow n = {\left( {{p^2} + 3p + 1} \right)^2} \\
 \]
From the above equation it is clear that as the number n can be written in the form of a square of some general variables.
So the given term n is a perfect square.
Hence, n will be odd as well as perfect square.
So, option A and C are the correct options.

Note:In order to solve these types of problems students must remember the general forms of some common number like consecutive numbers, odd numbers, even numbers etc. Also students must remember the method of completing squares in order to solve the general equations for checking if they are perfect squares or not.