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The Fibonacci numbers less than 100 are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and 89. The neighbors of each of these Fibonacci numbers, including the Fibonacci numbers that are neighbors of other Fibonacci numbers, are 0, 1, 2, 3, 4, 6, 7, 9, 12, 14, 20, 22, 33, 35, 54, 56, 88, and 90. We can eliminate the prime numbers because they have only two factors, which removes 2, 3, and 7. We can also remove 0 and 1. We can also remove 14 and 35 because when their digits are reversed, they are prime numbers with only two factors. The remaining numbers are 4, 6, 9, 12, 20, 22, 33, 54, 56, 88, and 90. Now, we check how many factors each of these numbers has:

The factors of 4 are 1, 2, and 4.

The factors of 6 are 1, 2, 3, and 6.

The factors of 9 are 1, 3, and 9.

The factors of 12 are 1, 2, 3, 4, 6, and 12.

The factors of 20 are 1, 2, 4, 5, 10, and 20.

The factors of 22 are 1, 2, 11, and 22.

The factors of 33 are 1, 3, 11, and 33.

The factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54.

The factors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56.

The factors of 88 are 1, 2, 4, 8, 11, 22, 44, and 88.

The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.

Three of these numbers 54, 56, and 88, have eight factors. If we reverse the digits of 88, we have the same number which has eight factors instead of four, so we can eliminate it. That leaves us with 54 and 56.

The factors of 45 are 1, 3, 5, 9, 15, and 45.

The factors of 65 are 1, 5, 13, and 65.

The only one with four factors is 65.

The answer is 56.

Q: I am a number less than 100. I have 8 factors including myself and 1. If you reverse my digits I'll only have 4 factors. I'm a neighbor of a Fibonaccci number. What am I?

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