Question

Which of the following is the greatest divisor of the product of any 3 consecutive even integers?

• A. 16
• B. 24
• C. 48
• D. 96

Answer 1

The Correct Option is C

Step 1: Consider any three consecutive even integers:

n, n + 2, n + 4 

Step 2: Factorize each term:

• Any even number n is divisible by 2.
• The next even number n + 2 is also divisible by 2.
• The third even number n + 4 is also divisible by 2.

Thus, their product n(n + 2)(n + 4) is always divisible by:

\[ 2 \times 2 \times 2 = 8 \]

Step 3: Check divisibility by 3:

• Among any three consecutive even numbers, one must be divisible by 4.
• Additionally, one of them is also divisible by 3.

This ensures that the product is divisible by:

\[ 8 \times 3 = 24 \]

Step 4: Check divisibility by higher numbers:

• Among three consecutive even numbers, one will be divisible by 2 again, making the divisibility:

\[ 24 \times 2 = 48 \]

Final Conclusion: The greatest divisor of the product of any three consecutive even numbers is 48. Thus, the correct answer is (C) 48.