Period fraction 966

It's well-known that if gcd $(10,N) = 1,$ the period of $\frac1{N}$ is the multiplicative order of $10$ mod $N$ , i.e. the smallest positive exponent $k$ such that $10^k \equiv 1$ mod $N.$ I'm not sure if there is an obvious way without brute force to enumerate the $N$ for which the multiplicative order of $10$ mod $N$ is $966,$ but a simple Python loop solved the problem immediately. I get the first five solutions to be $2021, 2303, 5969, 5977, 6063.$ So the answer is $\fbox{2021}.$

(If $N$ is not coprime to $10,$ there will be some digits before the repeating part, but the period will be the same as the period of $N/{\rm gcd}(10,N),$ so we get the minimal $N$ when that gcd is $1.$ )