The largest natural number by which the product of three consecutive even natural numbers is always divisible is _______.

• A. 16
• B. 24
• C. 48
• D. 96

The product of three consecutive even natural numbers can be represented as \( n(n+2)(n+4) \), where \( n \) is an even number. To determine the largest natural number that always divides this product, let's analyze the factors involved.

Understanding the Product

Three consecutive even numbers can be expressed as:

• First number: \( n \)
• Second number: \( n + 2 \)
• Third number: \( n + 4 \)

Factor Analysis

Each of these numbers is even, meaning they can be expressed as \( 2k \) for some integer \( k \). Therefore, the product is:

Product = n(n + 2)(n + 4) = 2k(2k + 2)(2k + 4)

Divisibility by 8

Among the three numbers, at least one of them is divisible by 4, and since they are all even, at least two of them contribute a factor of 2. Thus, the product is divisible by:

8 (from 4 and 2)

Divisibility by 3

Among any three consecutive even numbers, one of them must also be divisible by 3. This means the product is divisible by:

3

Combining Factors

Now, we combine the factors:

8 (from the even numbers) × 3 = 24

Conclusion

The largest natural number by which the product of three consecutive even natural numbers is always divisible is:

24

Thus, the correct answer is B. 24.