Giving you the product of the five numbers is equivalent to telling you the product of the two numbers I didn't choose. The only possible products that are achieved by more than one pair of numbers are $12(\{3,4\}$ and $\{2,6\})$ and $6(\{1,6\}$ and $\{2,3\})$. But in the second case, you at least know that the two unchosen numbers have odd sum (and so the five chosen numbers have odd sum also). Therefore, the first case must hold, and the product of the five chosen numbers is $$1 \cdot 2 \cdot 5 \cdot 6 \cdot 7=1 \cdot 3 \cdot 4 \cdot 5 \cdot 7=420$$