To find the largest natural number by which the expression n(n+1)(n+2) is divisible, we need to analyze the prime factors of the given expression. Since n is an even natural number, it can be expressed as n = 2k, where k is a natural number.


Let's substitute n = 2k in the given expression:
n(n+1)(n+2) = (2k)(2k+1)(2k+2)
= 2k(2k+1)(2k+2)

Now, let's consider the prime factorization of the expression:
2k(2k+1)(2k+2) = 2 * k * (2k+1) * (2k+2)


To find the largest natural number by which the expression is divisible, we need to identify the common prime factors. Let's analyze each term separately:

Term 1: 2
Term 2: k
Term 3: (2k+1)
Term 4: (2k+2)


The largest natural number by which the expression is divisible can be found by taking the product of the highest powers of the common prime factors.

Common Prime Factors: 2 and k

Since k is a natural number, it does not have any prime factors.


Therefore, the largest natural number by which the expression n(n+1)(n+2) is divisible is 2. This is because 2 is a common prime factor in all terms of the expression.