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### Question 12:
**"On dividing a number by 7, the remainder obtained is 4. What will be the remainder when the square of this number is divided by 7?"**

Let the number be \( x \). According to the problem, we have:

\[
x \equiv 4 \ (\text{mod} \ 7)
\]

This means that when \( x \) is divided by 7, the remainder is 4. Now, we need to find the remainder when \( x^2 \) is divided by 7. Squaring both sides of the congruence relation:

\[
x^2 \equiv 4^2 \ (\text{mod} \ 7)
\]

\[
x^2 \equiv 16 \ (\text{mod} \ 7)
\]

Now, divide 16 by 7:

\[
16 \div 7 = 2 \text{ remainder } 2
\]

So, the remainder when \( x^2 \) is divided by 7 is 2.

**Answer: 2**

---

### Question 13:
**"If the sum of a number and its square is 380, what is the number?"**

Let the number be \( x \). According to the problem, we have the equation:

\[
x + x^2 = 380
\]

Rearrange it into a quadratic equation:

\[
x^2 + x - 380 = 0
\]

Now, we solve this quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \( x^2 + x - 380 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = 1 \)
- \( c = -380 \)

Substitute these values into the quadratic formula:

\[
x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-380)}}{2(1)}
\]
\[
x = \frac{-1 \pm \sqrt{1 + 1520}}{2}
\]
\[
x = \frac{-1 \pm \sqrt{1521}}{2}
\]
\[
x = \frac{-1 \pm 39}{2}
\]

So, the two possible solutions are:

\[
x = \frac{-1 + 39}{2} = \frac{38}{2} = 19
\]
\[
x = \frac{-1 - 39}{2} = \frac{-40}{2} = -20
\]

Since we're likely looking for a positive number, the solution is \( x = 19 \).

**Answer: 19**

---

Do you need further details or have any questions?  
Here are 5 related questions for practice:

1. If a number leaves a remainder of 5 when divided by 6, what is the remainder when its square is divided by 6?
2. Solve \( x^2 + 2x - 15 = 0 \).
3. A number when divided by 9 leaves a remainder of 6. What will be the remainder when its cube is divided by 9?
4. If \( x + x^2 = 500 \), what is \( x \)?
5. What are the solutions to \( x^2 + 5x + 6 = 0 \)?

**Tip:** When solving quadratic equations, always check if you can factorize it before using the quadratic formula—it can save time!

On dividing a number by 7, remainder obtained is 4. What will be the remainder when the square of this number is divided by 7?

Find out how to calculate the remainder when the square of a number divided by 7 gives a remainder of 4. This question explores modular arithmetic and the properties of numbers. Suitable for Grades 8-10.

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