At an annual function of a school each student gives gift to every other student. If the number of gifts is 1980, find the number of students.

Answer

To solve the problem, we need to determine the number of students (let's denote it as \( x \)) given that each student gives a gift to every other student and the total number of gifts is 1980.

### Step-by-step Solution:

1. **Understanding the Problem**:
Each student gives a gift to every other student. Therefore, if there are \( x \) students, each student gives \( x - 1 \) gifts (since they do not give a gift to themselves).

2. **Setting Up the Equation**:
The total number of gifts given can be expressed as:
\[ \text{Total gifts} = x \times (x - 1) \]
We know from the problem that this total is 1980. Thus, we can write the equation:
\[ x(x - 1) = 1980 \]

3. **Rearranging the Equation**:
Rearranging the equation gives us:
\[ x^2 - x - 1980 = 0 \]

4. **Factoring the Quadratic Equation**:
We need to factor the quadratic equation \( x^2 - x - 1980 = 0 \). We look for two numbers that multiply to -1980 and add to -1. After checking the factors of 1980, we find:
\[ (x - 45)(x + 44) = 0 \]

5. **Finding the Roots**:
Setting each factor to zero gives us:
\[ x - 45 = 0 \quad \text{or} \quad x + 44 = 0 \]
This results in:
\[ x = 45 \quad \text{or} \quad x = -44 \]

6. **Choosing the Valid Solution**:
Since the number of students cannot be negative, we discard \( x = -44 \). Therefore, the number of students is:
\[ x = 45 \]

### Final Answer:
The number of students is **45**.