The answer is three significant figures. The 1 and the 7 are both significant, because they are non-zero quantities. This is where significant figures gets a little more complicated, because if a zero is used as a placeholder (i.e. 0.00027 cm) then it is insignificant. But in the case above, the zero isn't being used as a placeholder, and thus, is significant.
Therefore, three consecutive integers that add up to 312 are 103, 104, and 105. We know our answer is correct because 103 + 104 + 105 equals 312 as displayed above