Among the numbers 712, 1262, and 1680, which is the only one that can be written as a product of four consecutive natural numbers?
First, note that if three numbers are consecutive, then one of them is divisible by 3. Therefore, any number that is the product of three or more consecutive numbers must be divisible by 3. However, among the given numbers, only 1680 is divisible by 3, and furthermore,

$$
1680=2^{4} \times 3 \times 5 \times 7=5 \times 6 \times 7 \times 8
$$

Thus, 1680 is the only one among the three given numbers that can be written as the product of four consecutive numbers.