On dividing a number by 5, we get 3 as remainder. What will the remainder when the square of the this number is divided by 5?
A) 0

C) 2


B) 1

D) 4

E) None of these

Let the number be \( x \). According to the problem, when \( x \) is divided by 5, the remainder is 3. This can be written as:

\[
x \equiv 3 \ (\text{mod} \ 5)
\]

Now, we want to find the remainder when \( x^2 \) is divided by 5. To do this, square both sides of the congruence:

\[
x^2 \equiv 3^2 \ (\text{mod} \ 5)
\]

\[
x^2 \equiv 9 \ (\text{mod} \ 5)
\]

Now, divide 9 by 5 and find the remainder:

\[
9 \div 5 = 1 \text{ remainder } 4
\]

Thus, the remainder when \( x^2 \) is divided by 5 is 4.

The correct answer is:
**D) 4**

Would you like more details or have any questions? 

Here are 5 related questions for practice:
1. What will be the remainder when \( x^3 \) is divided by 5?
2. If a number leaves a remainder of 4 when divided by 7, what will be the remainder when its square is divided by 7?
3. Find the remainder when the cube of a number, which gives remainder 2 when divided by 5, is divided by 5.
4. What is the remainder when \( 15^2 \) is divided by 5?
5. If a number leaves remainder 1 when divided by 6, what will be the remainder when its square is divided by 6?

**Tip:** When working with remainders and squares, it can help to test small numbers and look for patterns!

On dividing a number by 5, we get 3 as remainder. What will the remainder when the square of this number is divided by 5? A) 0 B) 1 C) 2 D) 4 E) None of these

Solve the problem of finding the remainder when the square of a number, which gives remainder 3 when divided by 5, is divided by 5. This solution covers key number theory concepts like modular arithmetic and the properties of remainders. Suitable for Grades 9-10.

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