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5. Theorem: Prove that for any integer n, at least one of the integers n,
n+2, n+4 is divisible by 3.
If 3|n, then the theorem is true.
If ¬(3|n), then there is a remainder. There are two possible remainders
from division by 3: 1 and 2.
If the remainder of n/3 is 2, then 3|(n+2). The theorem is true in this
case.
If the remainder of n/3 is 1, then 3|(n+1). Since 3|n+4 iff 3|n+1, 3|(n+4)
also makes the theorem true.


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