solution: First we find the prime factorization of 288 to be $2^5\cdot 3^2$, and we have to split these factors among two consecutive even integers. The 3 must be with at least one 2 for the integer to be even, meaning one of the factors must be a multiple of $6.$ After some playing around, we find that when one factor is 18, that leaves us with $2^4=16$. So, our two integers are 16 and 18, with the greater integer being $\boxed{18}$.
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  The product of two consecutive positive even integers is 288. What is the greater of the two integers?
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 18
extra_info.question: The product of two consecutive positive even integers is 288. What is the greater of the two integers?
extra_info.split: test