Cab number leaving remainder one and divisible by 11

A rude cab driver gives his number as a puzzle: divided by any of $2,3,4,5,6$ it leaves remainder $1$, it is a multiple of $11$, and it is the smallest positive number with both properties. What is it?

$121$. The number is $1$ more than a multiple of $\operatorname{lcm}(2,3,4,5,6)=60$: $1,61,121,\dots$, and the first of these divisible by $11$ is $121$.