@Caerus
Let k be any integer.
Theorem: Either k, or k + 2, or k + 4 is divisible by 3.
Proof. Suppose, to the contrary, that none is divisible by 3. Then certainly k is not divisible by 3. This leads to 2 cases:
Case 1: k = m + 1, where m is an integer. In this case, k + 2 = (m + 1) + 2 = m + 3 = 3(m + 1). This makes k + 2 a triple of an integer, thus k + 2 is divisible by 3, a contradiction.
Case 2: k = m + 2, where m is an integer. In this case, k + 4 = (m + 2) + 4 = m + 6 = 3(m + 2). This makes k + 4 a triple of an integer, thus k + 4 is divisible by 3, a contradiction.
In either case, we get a contradiction. Thus, the assumption that none is divisible by 3 can not be correct. Hence, at least one of them is divisible by 3.
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