Let's break down the problem:

Given:

• A number $n$ when divided by 5 gives a remainder of 3. This means we can express $n$ as: $n = 5k + 3$ for some integer $k$.

We need to find the remainder when the square of this number ($n^2$) is divided by 5.

Step-by-step Solution:

1. Since $n = 5k + 3$, let's square $n$: $n^2 = (5k + 3)^2 = 25k^2 + 30k + 9$

2. Simplify $n^2$ modulo 5:

   • Notice that each term can be reduced modulo 5:
     • $25k^2 \equiv 0$ (since 25 is a multiple of 5)
     • $30k \equiv 0$ (since 30 is also a multiple of 5)
     • $9 \equiv 4$ (since 9 modulo 5 gives a remainder of 4)

   Therefore, $n^2 \equiv 4 \mod 5$

Conclusion:

The remainder when the square of this number is divided by 5 is 4.

Thus, the correct answer is: $\boxed{4} \quad (\text{Option D})$

Would you like further explanation or have any other questions?

Here are some related questions to deepen your understanding:

1. What is the general formula for finding the remainder when squaring a number with a specific remainder?
2. How would the remainder change if the original number had a remainder of 2 when divided by 5?
3. If $n = 5k + 1$, what would be the remainder when $n^2$ is divided by 5?
4. What patterns can be observed in the squares of numbers modulo 5?
5. How would this problem change if we used a different divisor instead of 5?

Tip: When solving problems involving squares and remainders, it's helpful to express the number in a modular form and then simplify the expression step-by-step.