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dgentile
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Homework Statement
Prove that 3 divides one of the integers n, n + 2, or n + 4, for any integer n.
Proving that 3 divides at least one integer means showing that there exists an integer that can be evenly divided by 3 without leaving a remainder.
This is an important concept in number theory and mathematics in general. It helps us understand divisibility and the properties of integers, and also has applications in various other fields such as cryptography and computer science.
The most common way to prove this is by using a direct proof. This involves assuming that there is no integer that can be divided by 3 without a remainder, and then arriving at a contradiction. This shows that our initial assumption was incorrect and there must be at least one integer that 3 can divide evenly.
Sure, for example, to prove that 3 divides at least one integer in the set {2, 5, 7, 9}, we can use a direct proof. We assume that no integer in this set can be divided by 3 without a remainder. However, 9 can be divided by 3 to give 3 as a quotient. This contradicts our initial assumption, proving that 3 does indeed divide at least one integer in this set.
Yes, there are several other methods such as using modular arithmetic, proof by contradiction, and proof by contrapositive. However, the direct proof method is the most commonly used and straightforward method for proving this statement.