Let α 1 be a root of z^p+q-z^p-z^q+1=0, p,q distinct primes. Show that either 1+α+·s+α^p-1=0 or 1+α+·s+α^q-1=0, but not both.

Answer: Proved. z^p+q-z^p-z^q+1=(z^p-1)(z^q-1). So α^p=1 or α^q=1; since α 1 this gives the p- or q-sum =0. Both would force α to be a common p- and q-th root =1 (as gcd(p,q)=1), which is excluded. JEE Advanced 2002 · Complex Numbers — verified solution by the Vidaara Team.

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[JEE Advanced 2002] let alpha 1 be a root of z p q z p z q

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Let $\alpha\ne1$ be a root of $z^{p+q}-z^p-z^q+1=0$, $p,q$ distinct primes. Show that either $1+\alpha+\cdots+\alpha^{p-1}=0$ or $1+\alpha+\cdots+\alpha^{q-1}=0$, but not both.

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VAVidaara Admin ✓ Vidaara Team ✓ Accepted · 2d ago ▲ 0

Answer: Proved.

$z^{p+q}-z^p-z^q+1=(z^p-1)(z^q-1)$. So $\alpha^p=1$ or $\alpha^q=1$; since $\alpha\ne1$ this gives the $p$- or $q$-sum $=0$. Both would force $\alpha$ to be a common $p$- and $q$-th root $=1$ (as $\gcd(p,q)=1$), which is excluded.

JEE Advanced 2002 · Complex Numbers — verified solution by the Vidaara Team.

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