why if x is an intiger does y always finish with 0 where y=x+x^2+x^3+x^4?

Answer 1

and

x = 6 ; y = 6 + 36 + 216 + 1296 = 1554

x = 11; y = 11 + 121 + 1331 + 14641 = 16104

x = 16 y = 69904

x = 21 y = 204204

but it works for the ones in between

i.e. it works where x = 5n, 5n + 2, 5n + 3 5n + 4 but not 5n + 1

factoring x+x^2+x^3+x^4 = x + x^3 + x^2 + x^4
= x (1 + x^2) + x^2 (1 + x^2) = x (1 + x) (1 + x^2)

5n is a multiple of 5 and 5n + 1 is even
with 5n + 2 and 5n + 3, (1 + x^2) is a multiple of 5 and either x or x + 1 is even
with 5n + 4, x + 1 = 5n + 4 + 1 - a multiple of 5; again x or x + 1 is even

only with 5n + 1 is there no multiple of 5 involved

Answer 2

I don't think so, unless there is more to your question.

Just substitute x =1 which is an integer. You get y= 4.

Answer 3

because y is a different number.