The number of natural numbers n in the interval `[1005, 2010]` for which the polynomial `1+x+x^(2) + x^(3) + "….."x^(n-1)` divides the polynomial `1 + x^(2) + x^(4) + x^(6) + "……"x^(2010)` is -

Options

A. `0`
B. `100`
C. `503`
D. `1006`

Answer

C

Explanation

`1 + x^(2) + x^(4)"....."x^(210) = (1(1-x^(210)))/(1-x^(2)) = ((1-x^(1006))(1+x^(1006)))/((1-x)(1+x))`
`= (1+x^(1006))(((1-x^(503)))/((1-x)))(((1+x^(503)))/((1+x)))`
`=(1+x^(1006))(1+x+x^(2)+"...."x^(502))(1-x+x^(2)-x^(3)+"....."x^(502))`
that is divisible by `1+ x + x^(2) + "...."x^(n-1)`
If ` n - 1 = 502`
`n = 503`