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A dodecahedral number can never be prime--the nth dodecahedral number is always divisible by n. This can be proved very simply:
- n is part of the numerator. There are no fractions in the numerator alone, so the numerator is divisible by n.
- Out of or , one of the two must be even. Therefore, the numerator is divisible by 2.
- Given the above, the numerator must be divisible by 2n.
- Noting the denominator, . Therefore, the nth dodecahedral number is always divisible by n.
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