The product of two successive natural numbers is 1980. Which is the smaller number?

Options

A. 34
B. 35
C. 44
D. 45

Answer

To find the smaller of two successive natural numbers whose product is 1980, we can follow these steps:

### Step 1: Define the natural numbers
Let the first natural number be \( x \). Then, the next successive natural number will be \( x + 1 \).

### Step 2: Set up the equation
The product of these two numbers can be expressed as:
\[ x(x + 1) = 1980 \]

### Step 3: Expand the equation
Expanding the left side gives:
\[ x^2 + x = 1980 \]

### Step 4: Rearrange the equation
Rearranging the equation to set it to zero:
\[ x^2 + x - 1980 = 0 \]

### Step 5: Factor the quadratic equation
We need to factor the quadratic equation \( x^2 + x - 1980 = 0 \). We are looking for two numbers that multiply to \(-1980\) and add to \(1\).

### Step 6: Find the factors of 1980
To find the factors of 1980, we can start by factoring it:
\[ 1980 = 2 \times 990 = 2 \times 2 \times 495 = 2^2 \times 3^2 \times 5 \times 11 \]
From this, we can find pairs of factors that could potentially add up to \(1\).

### Step 7: Identify the correct factors
After testing various pairs, we find that:
\[ 44 \text{ and } -45 \]
are the factors we need since:
\[ 44 \times (-45) = -1980 \quad \text{and} \quad 44 + (-45) = -1 \]

### Step 8: Write the factored form
Thus, we can express the quadratic as:
\[ (x - 44)(x + 45) = 0 \]

### Step 9: Solve for \( x \)
Setting each factor to zero gives us:
\[ x - 44 = 0 \quad \Rightarrow \quad x = 44 \]
\[ x + 45 = 0 \quad \Rightarrow \quad x = -45 \quad (\text{not a natural number}) \]

### Step 10: Conclusion
Since \( x = 44 \) is the only valid solution, the smaller number is:
\[ \text{Smaller number} = 44 \]