Finding the Greatest Divisor for Consecutive Even Numbers Product

Let the three consecutive even numbers be represented algebraically as $N$, $N+2$, and $N+4$. Their product is $P = N \times (N+2) \times (N+4)$. The question asks for the greatest number from the given options that exactly divides such a product $P$. We need to find the largest value among {12, 24, 48, 64} that is a possible divisor for $P$. We will test this by checking specific examples.

Testing Divisibility with Examples

Let's test a few sets of three consecutive even numbers:

• Set 1: The numbers are 2, 4, 6. The product $P = 2 \times 4 \times 6 = 48$. Checking divisibility by the options:
  • $48 \div 12 = 4$
  • $48 \div 24 = 2$
  • $48 \div 48 = 1$
  • $48 \div 64$ is not an integer.
  In this case, 64 does not divide the product.
• Set 2: The numbers are 4, 6, 8. The product $P = 4 \times 6 \times 8 = 192$. Checking divisibility by the options:
  • $192 \div 12 = 16$
  • $192 \div 24 = 8$
  • $192 \div 48 = 4$
  • $192 \div 64 = 3$
  In this case, 192 is divisible by 12, 24, 48, and 64.

We are looking for the greatest number among the options {12, 24, 48, 64} that *can* divide the product. From the examples:

• 12 divides 48 and 192.
• 24 divides 48 and 192.
• 48 divides 48 and 192.
• 64 divides 192 (but not 48).

Since 64 is an option and it successfully divides the product in the second example (192), and it is the largest value among the options that demonstrates this capability for at least one case, it satisfies the condition.

Conclusion

Comparing the options {12, 24, 48, 64}, the greatest number is 64. This number exactly divides the product for the set {4, 6, 8}.