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Algebra

Scorching goIITian

Joined:	18 Feb 2012
Post:	264

2 Apr 2012 15:13:50 IST

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300

'COMPLEX' complex no....

Engineering Entrance , JEE Main , JEE Advanced , Mathematics , Algebra

x=w^2009^2009^2009.......2009 times..... then 1+x+x3=??? w is cube root of unity

Comments (4)

Aman Gupta

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Joined: 10 Feb 2012

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2 Apr 2012 15:17:51 IST

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Tarun Garg

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Joined: 21 Mar 2012

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2 Apr 2012 15:29:21 IST

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it is 0

hemang

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Joined: 27 Dec 2010

Posts: 1508

3 Apr 2012 10:34:47 IST

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see, 2009 is 2(mod 3).

and we raise it to an odd power,

2(mod 3) remains the same.

lemma -

2^(2n-1) is 2(mod 3) for all natural numbers , n.

4^n - 2 ,

(3+1)^n - 2.

(multiple of 3) - 1.

which is 2(mod 3). :)

so x = w^(power that is 2(mod3)).

x = w^2.

we want 1 + x + x^3 = 1 + w^2 + w^6 = 2 + w^2 = 1 + 1 + w + w^2 - w = 1 - w.

not sure. ;(

kartikAIR1

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Joined: 2 Mar 2012

Posts: 304

3 Apr 2012 11:41:00 IST

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I agree with hemang !! x = w^2 !! sure..

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