So, you are looking for such that . So, suppose (it can be 2 or 5). Then , so . That gives , and a carry of 1. So, . This gives as you suggested. Again, the carry is 1. Then, , so and the carry is 3. Next, . So, , and the carry is 5. So, I don't know how you calculated the square root of 305, but I got a very different result.