You are Intelligent

If anyone knows even slightly about remainder, his chance of answering this question correctly will be $100\%$ . Only three possible options of remainder: $0,1$ & $2$ . And guess what? You have three chances. It would have been better if the question was posted with multiple choice option. However, here we go:

$\color{#0C6AC7} {\text{The divisibility rule of 3 is, "If we add the digits of the dividend to get}}$ $\color{#0C6AC7} {\text{a new number \& divide it by 3, the remainder will still be the same." }}$

${9+0+6+2+7+1+1+5+9+0+1+6+7+1=55\, \equiv 1\pmod3}$ ${\Rightarrow 5+5=10 \,\equiv 1\pmod3}$ ${\Rightarrow 1+0=1 \, \equiv 1\pmod3}$

We can get the answer directly from the first line as remainder of $55$ is $1$ , when divided by $3$ . Whatsoever, following the divisibility rule of $3$ , we get the remainder $\color{#69047E} {\boxed 1}$ .

I need some mental calculations in here :)

First, we add the digits together:

$9+0+6+2+7+1+1+5+9+0+1+6+7+1 = 55$

Then, $5+5 =10$

Then, $1+0 = 1$ . Thus, this number is not divisible by $3$ . Divide $1$ by $3$ we get $\boxed{1}$

It's either $0,1$ or $2$ ... Easiest way check the three you will get it(only for those lazy like me)

If you make each digit add up to a multiple of three, you can whether or not it will have a remainder without add them all up. And since one is left over, it means that the remainder would be one as the number would be a multiple of 3 without it