solution: From $3m + 4n = 100$ we can get that $n = 25 - \frac{3}{4}m$.  Then we want to minimize $$\left| m - n \right| = \left| m - 25 + \frac{3}{4}m \right| =\left|  \frac{7}{4}m - 25 \right| = \left|  7m - 100 \right|$$In other words we want $7m$ as close to $100$ as possible while still giving us integer solutions for $m$ and $n$ to the equation $3m + 4n = 100$.
  
  By trial and error, we can find that the solution to $3m + 4n = 100$ that makes $m$ closest to $\frac{100}{7}$ is $(m,n) = (16,13)$.  Then we have $\left| m - n \right| = 16-13 =\boxed{3}$.
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  If $m$ and $n$ are integers such that $3m + 4n = 100$, what is the smallest possible value of $\left| m - n \right|$ ?
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 3
extra_info.question: If $m$ and $n$ are integers such that $3m + 4n = 100$, what is the smallest possible value of $\left| m - n \right|$ ?
extra_info.split: test