The sum of four integers in A.P. is 24 and their product is 945. Find the product of the smallest and greatest integers :

Options

A. 30
B. 27
C. 35
D. 39

Answer

To solve the problem step by step, we need to find four integers in Arithmetic Progression (A.P.) such that their sum is 24 and their product is 945. We will also find the product of the smallest and greatest integers among them.

### Step 1: Understanding the A.P. Structure
Let the four integers in A.P. be represented as:
- First term: \( a - 3d \)
- Second term: \( a - d \)
- Third term: \( a + d \)
- Fourth term: \( a + 3d \)

Here, \( a \) is the middle value and \( d \) is the common difference.

### Step 2: Setting Up the Sum Equation
The sum of these four integers can be expressed as:
\[ (a - 3d) + (a - d) + (a + d) + (a + 3d) = 4a \]
Given that their sum is 24:
\[ 4a = 24 \]
Solving for \( a \):
\[ a = \frac{24}{4} = 6 \]

### Step 3: Setting Up the Product Equation
Now, we need to find the product of these integers:
\[ (a - 3d)(a - d)(a + d)(a + 3d) \]
Substituting \( a = 6 \):
\[ (6 - 3d)(6 - d)(6 + d)(6 + 3d) \]
This expression can be simplified using the difference of squares:
\[ [(6 - 3d)(6 + 3d)][(6 - d)(6 + d)] = (36 - 9d^2)(36 - d^2) \]
We know that this product equals 945:
\[ (36 - 9d^2)(36 - d^2) = 945 \]

### Step 4: Expanding the Product
Let \( x = 9d^2 \). Then, we can rewrite the equation:
\[ (36 - x)(36 - \frac{x}{9}) = 945 \]
Expanding this gives:
\[ (36 - x)(36 - \frac{x}{9}) = 36^2 - 36 \cdot \frac{x}{9} - 36x + \frac{x^2}{9} = 945 \]

### Step 5: Solving for \( d \)
After simplifying and solving the quadratic equation, we can find possible values for \( d \). However, we can also directly find integer factors of 945 that could fit the A.P. structure.

### Step 6: Finding Integer Factors
The prime factorization of 945 is:
\[ 945 = 3^3 \times 5 \times 7 \]
We can try combinations of these factors to find integers that sum to 24.

### Step 7: Testing Combinations
After testing various combinations, we find:
- \( 3, 5, 7, 9 \) are in A.P. (with common difference 2).
- Their sum is \( 3 + 5 + 7 + 9 = 24 \).
- Their product is \( 3 \times 5 \times 7 \times 9 = 945 \).

### Step 8: Finding the Product of Smallest and Greatest
The smallest integer is \( 3 \) and the greatest integer is \( 9 \). Therefore, the product of the smallest and greatest integers is:
\[ 3 \times 9 = 27 \]

### Final Answer
The product of the smallest and greatest integers is \( 27 \).

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