What is the property of the square of any integer, and what is the square root of 1681?

a) The square of any integer is always:
Let's consider examples:

• If the integer is positive, e.g., 2, then $2^2 = 4$, which is positive.
• If the integer is negative, e.g., $-3$, then $(-3)^2 = 9$, which is positive.
• If the integer is 0, then $0^2 = 0$. So, the square of any integer is always either positive or zero (non-negative). Among the given options, "positive" is the most general characteristic for non-zero integers, and squares are never negative. If "non-negative" were an option, it would be more precise. Given the choices, "positive" is the intended answer for the general case, acknowledging that $0^2=0$. The correct option is ii).

$$\boxed{\text{ii) positive}}$$

b) The square root of 1681 is:
To find the square root of 1681, we can test the given options:

• i) $31^2 = 31 \times 31 = 961$
• ii) $41^2 = 41 \times 41 = 1681$
• iii) $45^2 = 45 \times 45 = 2025$ The square root of 1681 is 41. The correct option is ii).

$$\boxed{\text{ii) } 41}$$