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First, one must acknowledge that if one adds a group of numbers, the resulting sum will have the same remainder when divided by three as the sum of each number's modulus by three would be. The first four numbers in the Fibonacci sequence, mod 3, are 0, 1, 1, 2. By the rule established in the beginning of this answer and by the rules of the Fibonacci sequence, the modulus of a given number in the sequence is equal to the sum of the previous two, mod 3. As a result, the next four numbers are 0, 2, 2, 1. The rule holds true thus far, with the first number in each set of modulus figures being 0. After that, the next four numbers are 0, 1, 1, 2. As a result, it can be seen that the Fibonacci sequence, modulus 3, has the following patter, which shows that each fourth number is divisible by three: 0, 1, 1, 2, 0, 2, 2, 1.

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Q: Prove 3 divides every 4th Fibonacci number?
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