The number of natural numbers n in the interval [1005,2010] for which the polynomial 1+x+x²+x³+ . x^ n-1 divides the polynomial 1+x²+x³+x⁴+ .+x²⁰¹⁰ is-

Step-by-step solution

array l 1+x²+x⁴+ x²⁰¹⁰= 1 (1-x²⁰¹² ) 1-x² = (1-x¹⁰⁰⁶ ) (1+x¹⁰⁰⁶ ) (1-x)(1+x) = (1+x¹⁰⁰⁶ ) ( (1-x⁵⁰³ ) (1-x) ) ( (1+x⁵⁰³ ) (1+x) ) = (1+x¹⁰⁰⁶ ) (1+x+x²+ . x⁵⁰² ) (1-x+x²-x³+ . x⁵⁰² ) array this is divisible by 1+x+x²+ . . x^ n-1 if n-1=502 n=503