Introduction:
We are given that set K consists of even natural numbers. We need to find the largest natural number that will completely divide the product of any 3 consecutive numbers from set K.

Analysis:
Let's analyze the factors that will divide the product of any 3 consecutive numbers from set K.

Even Numbers:
Since set K consists of even natural numbers, any 3 consecutive numbers from set K will also be even.

Factors of Even Numbers:
Every even number can be expressed as the product of 2 and another number. The factors of an even number will always be 1, 2, and the number itself.

Prime Factors:
To find the largest natural number that will completely divide the product of any 3 consecutive numbers from set K, we need to find the common prime factors of these 3 consecutive numbers.

Maximum Prime Factors:
Since the numbers in set K are consecutive even numbers, their prime factors will be the same. The maximum prime factor of any number in set K will be the same for all numbers in the set.

Evaluating the Maximum Prime Factor:
Let's consider an example to evaluate the maximum prime factor. Suppose we have 3 consecutive even numbers from set K: 2, 4, and 6.

The prime factors of 2 are {2}.
The prime factors of 4 are {2, 2}.
The prime factors of 6 are {2, 3}.

The common prime factors of these 3 numbers are {2}. Therefore, the maximum prime factor is 2.

Calculating the Largest Natural Number:
To calculate the largest natural number that will completely divide the product of any 3 consecutive numbers from set K, we need to find the maximum prime factor of any number in set K and multiply it with the next prime number.

Largest Natural Number:
The largest prime factor of any number in set K is 2. The next prime number is 3.

The largest natural number = 2 * 3 = 6.

Conclusion:
The largest natural number that will completely divide the product of any 3 consecutive numbers from set K is 6.