Let $x^2+x+1$ is divisible by $3$. If $x$ is divided by $3$, the remainder will be

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Solution

Given: $x^2+x+1$ is divisible by $3$.

Case $I:$ $x=3m$

$\therefore 9m^2+3m+1$

$\therefore$ Not divisible by $3$.

Case $II:$ $x=3m+1$

$\therefore 9m^2+9m+3$

$\therefore$ Divisible by $3$.

Case $III:$ $x=3m+2$

$\therefore 9m^2+15m+7$

$\therefore$ Not divisible by $3$.

If $x^2+x+1$ is divisible by $3$, $x=3m+1$

$\therefore$ Remainder $=1$

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