solution: The number $21420N$ is divisible by $6$ if and only if it is even and the sum of its digits is divisible by $3$. So $N$ must be even, and $2 + 1 + 4 + 2 + 0 + N = 9 + N$ must be divisible by $3$.  Since $9$ is divisible by $3$, we see that $N$ must also be divisible by $3$. The only digit that works is $N = \boxed{6}$.
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  Let $N$ be the units digit of the number $21420N$. Which nonzero value of $N$ makes this number divisible by $6$?
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 6
extra_info.question: Let $N$ be the units digit of the number $21420N$. Which nonzero value of $N$ makes this number divisible by $6$?
extra_info.split: test