ISI 2004 MIII

Question

$x^{2}+x+1$ is a factor of $\left ( x+1 \right )^{n}-x^{n}-1$ whenever

• A.  $n$ is odd
• B. $n$ is odd and multiple of $3$
• C. $n$ is an even multiple of $3$
• D. $n$ is odd and not a multiple of $3$

Answer 1

Keep $n=3$ it would come out to be $3x^2 +3x ,$ so it would be a factor of $x^2 +x+1$

Comments

Therefore answer should be c

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@kaluti,how 3*(x^(2)) +3*(x) is a multiple of x^(2) +(x)+1?would'nt 1 be a remainder then??

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Answer will be (D)

https://math.stackexchange.com/questions/3150295/x2x1-is-a-factor-of-x1n-xn-1-whenever?noredirect=1&lq=1

https://math.stackexchange.com/questions/1237256/find-n-such-that-x2-x-1-is-a-factor-of-x1n-xn-1

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If w is a cube root of unity,  then w is a root of1+x+x^2=0.

Now x+1=-x^2

Then if n is odd and not multiple of three, (-x^2)^n+-x^n-1=-(x^n)^2-x^n-1=0

So correct answer is D