7. (10 points) There are some natural numbers, each of which, when multiplied by 7, has a product whose last four digits are 2012. Among these natural numbers, the smallest number is $\qquad$ .


【Analysis】First, analyze the problem by solving it in reverse; to find the smallest multiplier, it can be converted to finding the smallest product. 2012 is not a multiple of 7, so we need to add the smallest possible digit at the front to make it a multiple of 7.
【Solution】Solution: First, notice that 2012 is not a multiple of 7, so to find the smallest number, we need to check what happens when we add the smallest digit at the front to form a multiple of 7.

Adding the digit $1$ at the front, $12012 \div 7 = 1716$.