Number Theory

We have a system of equations to solve. We have

$N \equiv 2 \pmod{3}, \\ N \equiv 2 \pmod{7}, \\ N \equiv 2 \pmod{10}.$

Now, let's note something really important. All 3 equations have a remainder of 2. 

Let's anakyze. 

N leaves a remainder of 2 when divided by 3, 7, and 10. 

Therefore, our answer is the LCM of 3, 7, and 10. 

This is because since they all leave the same remainder, the same number must be for all of them. 

Therefore, we have $LCM[3, 7, 10] = 210$

Adding 2, we have $210+2=212$

So our answer is 212. 

Thanks! :)