Question

how to prove that the difference between squares of consecutive even numbers is always a multiple of 4

Answer 1

Let x be an even integer then x+2 is the next consecutive even integer.

We need to look at
`(x+2)^2 - x^2`, since that's the difference between the squares of any two consecutive even integers.

`(x+2)^2 - x^2 = x^2 + 4x + 4 - x^2`

`=4x+4`

`=4(x+1)`

Since that difference is 4 times something, it is a multiple of 4.