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^3 + x^4 + …. + x^2010 is-
(A) 0 (B) 100 (C) 503 (D) 1006

• (1+X^2+X^4+......+X^2010) ------------ POLYNOMIAL
  
  (1+X^2+X^4+......+X^2010) ------>THIS IS A GP SERIES..... LETS FACTORISE THIS
  = (1-(X^2)^1006))/(1-X^2) .......a(1-r^n)/(1-r)
  (1-(X^2012))/(1-X^2)
  
  
  (THIS IS OF THE THE FORM A^2-B^2=(A+B)(A-B))
  
  [(1-(X^1006))((1+(X^1006))]/[(1-X)(1+X) ]
  
  (WHICH IS AGAIN OF THE THE FORM A^2-B^2=(A+B)(A-B))
  
  [(1-(X^503)) (1+(X^503)) ((1+(X)^1006))]/[(1-X)(1+X) ]
  
  ....... FROM THIS WE GET 2 GP SERIES
  {
  i.e [(1-(X^503))/(1-X)] = (1+X+X^2+X^3+X^4+......+X^502)
  [(1+(X^503))/(1-X)] = (1-X+X^2-X^3+X^4+.....+X^502)
  }
  
  =[1+X^1006][(1+X+X^2+X^3+X^4+......+X^502) (1-X+X^2-X^3+X^4+......+X^502)]
  ^^^
  |||
  (1 + x + x^2 + x^3 + …. x^n–1) IS A FACTOR AS GIVEN IN QUESTION
  
  ( THEREFORE BY COMPARISON N=503)
• 5 years agoHelpfull: Yes(2) No(0)