SOLUTION: Prove with a simple equation that the product of four consecutive integers can never be a perfrct square.

Question 34722This question is from textbook
: Prove with a simple equation that the product of four consecutive integers can never be a perfrct square. This question is from textbook

Found 2 solutions by longjonsilver, venugopalramana:
Answer by longjonsilver(2297) (Show Source): You can put this solution on YOUR website!
not sure if this is a rigourous proof, but it should at least show you some things.

Take an example: 2*3*4*5. To be a perfect square, we need something like 3*3 or 7*7 or 15*15 etc.

So with 2*3*4*5, we need two of the integers to multiply to equal the other two integers. The only remote possibility of this is if 2*5=3*4, which it doesn't in this case, since 10 is not 12.

So, algebraically, x(x+1)(x+2)(x+3) are our four integers, starting at number "x".

If we then say, OK, for which value of x does x(x+3) = (x+1)(x+2) hold? then we can find the answer(s). From the wording of the question, we are looking for no solution.

which is patently not true, so there is no value of x such that is true.

jon.

Answer by venugopalramana(3286) (Show Source): You can put this solution on YOUR website!
LET THE 4 CONSECUTIVE INTEGERS BE N-1,N,N+1,N+2
THEIR PRODUCT = P =(N-1)N(N+1)(N+2)=(N^2-1)(N^2+2N)=N^4+2N^3-N^2-2N....IF THIS IS TO BE A PERFECT SQUARE THEN IT SHOULD BE IN THE FORM OF
(N^2+AN)^2...SINCE THERE IS NO CONSTANT TERM IN P...EXPANDING
(N^2+AN)^2=N^4+2AN^3+A^2N^2..EQUATING WITH P WE SHOULD HAVE
2A=1...OR...A=1/2
A^2=-1..AND -2N=0.....WHICH IS NOT POSSIBLE .....HENCE P CANNOT BE WRITTEN AS A PERFECT SQUARE.
RELATED QUESTIONS
Prove that the product of two consecutive nonzero even integers is never a perfect... (answered by Alan3354)
Prove that if 1 is added to the product of any four consecutive integers, the sum is a... (answered by ikleyn,Boreal,KMST)
Prove that the product of three consecutive integers is divisible by 6; of four... (answered by Edwin McCravy)
can you find the second of four consecutive even integers such that the product of the... (answered by stanbon)
prove that product of 4 consecutive numbers cannot be the square of an... (answered by Alan3354)
I'm really sorry, I don't study maths in English so I really have no idea where to put... (answered by CubeyThePenguin,ikleyn)
Four times the sum of three consecutive integers is 48 a. Write an equation that could... (answered by Alan3354)
Which of the following statements can be used to prove that the sum of the squares of 3... (answered by CubeyThePenguin)
Why is it that irrational numbers can never be expressed as a quotient of... (answered by stanbon,Tobiasz,edjones)