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

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 odd (if not we will do something similar with (n+1)). We can then see that n is relatively prime to n-1,n+1, 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+1)(n+2) musr also be a perfect cube. But multiplied out this is (n^2 - 1)(n+2) = n^3 +2n^2 - n - 2. 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

nicely, you advise effective integers 0*a million*2*3 = 0, this is a suitable dice of 0 EDIT No, organic start up from 0. Im sorry. Mathematicians use N or (an N in blackboard formidable) to consult from the set of all organic numbers. This set is infinite yet countable by skill of definition. To be unambiguous approximately whether 0 is secure or no longer, each and every so often an index "0" is further in the former case, and a superscript "*" is further in the latter case: N0 = { 0, a million, 2, ... } ; N* = { a million, 2, ... }. EDIT 2 sturdy answer, Ben. I did somethink like that yet I couldnt locate the thank you to coach that this product wasnt a suitable dice. thank you for this evidence. Ana

Answer 3

let the 4 numbers be x, x+1, x+2, x+3

Let the product P = x(x+1)(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 is not in the form of (x+a)^3
Thus not a perfect cube of any natural no.

Eg: let x = 1

thus P = 1 *2 * 3* 4 = 24 where 24 is not a perfect cube of any no.

Answer 4

We can do this very easily using the Principle of Mathematical Induction, which you might be knowing, I suppose.