The largest possible number by which the product of any five consecutive natural numbers can be divided :

Options

A. 120
B. 160
C. 100
D. none of these

Answer

To find the largest possible number by which the product of any five consecutive natural numbers can be divided, we can follow these steps:

### Step 1: Understand the Product of Five Consecutive Natural Numbers
Let’s denote five consecutive natural numbers as \( n, n+1, n+2, n+3, n+4 \). The product of these numbers can be expressed as:
\[ P = n \times (n+1) \times (n+2) \times (n+3) \times (n+4) \]

### Step 2: Calculate the Product for the Smallest Case
To find a specific example, we can calculate the product of the first five natural numbers:
\[ P = 1 \times 2 \times 3 \times 4 \times 5 \]
Calculating this gives:
\[ P = 1 \times 2 = 2 \]
\[ P = 2 \times 3 = 6 \]
\[ P = 6 \times 4 = 24 \]
\[ P = 24 \times 5 = 120 \]

### Step 3: Factorial Representation
The product of the first five natural numbers is known as \( 5! \) (5 factorial), which is equal to 120. Therefore, we can say:
\[ P = 5! = 120 \]

### Step 4: Generalize for Any Five Consecutive Natural Numbers
For any five consecutive natural numbers, the product will always include the factors of \( 5! \). This means that no matter which five consecutive numbers we choose, the product will always be divisible by 120.

### Step 5: Conclusion
Thus, the largest possible number by which the product of any five consecutive natural numbers can be divided is:
\[ \text{Answer: } 120 \]