Prove that the product of 4 consecutive natural numbers can never be a perfect cube of any natural number?

Question

Prove that the product of 4 consecutive natural numbers can never be a perfect cube of any natural number

To the first answerer:
Nice try, but natural numbers start from 1. Whole numbers start from 0. Therefore the entry of 0 is not satisfactory

Answer 1

I thought about this for a bit, but got impatient, so I Googled it.

A product of four consecutive positive integers can be written as: (n-1)n(n+1)(n+2). Suppose n is even (if not we will do something similar with n). We can then see that n+1 is relatively prime to n-1,n, and n+2. Why? Because any common factor of a and b must also divide the remainder when a is divided by b.

Since n is relatively prime to the other three numbers, n must be a perfect cube, so (n-1)n(n+2) must also be a perfect cube. But multiplied out this is n^3 + n^2 - 2n. This number cannot be a perfect cube, since it lies between two consecutive perfect cubes: n^3 and (n+1)^3 = n^3 + 3n^2 + 3n +1.

Thus the product of four consecutive positive integers cannot be a perfect cube.

Answer 2

Well, you mean positive integers

0*1*2*3 = 0, which is a perfect cube of 0

EDIT

No, natural start from 0. Im sorry.

Mathematicians use N or (an N in blackboard bold) to refer to the set of all natural numbers. This set is infinite but countable by definition.

To be unambiguous about whether zero is included or not, sometimes an index "0" is added in the former case, and a superscript "*" is added in the latter case:

N0 = { 0, 1, 2, ... } ; N* = { 1, 2, ... }.

EDIT 2

Good answer, Ben. I did somethink like that but I couldnt find a way to prove that this product wasnt a perfect cube. Thanks for this proof.


Ana

Answer 3

enable the 4 numbers be x, x+a million, x+2, x+3 enable the product P = x(x+a million)(x+2)(x+3) = (x^2 + x)(x^2+ 5x + 6) = (x^4 + 5x^3 + 6x^2)+ x^3+ 5x^2+ 6x = x^4 + 6x^3+11x^2+6x which isn't in the form of (x+a)^3 hence no longer a suitable dice of any organic no. Eg: enable x = a million hence P = a million *2 * 3* 4 = 24 the place 24 isn't a suitable dice of any no.