Latest KVPY Aptitude Question SOLUTION: 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

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(#M40027248) KVPY Logical Reasoning question Mental Ability -2010 Keep an EYE

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

Asked In KVPY (11 years ago)

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L.SUGANDA PRASAD  (6 years ago)

(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)

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