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I have identified few unique characteristics of fifth power of a number i.e. $n \cdot n \cdot n \cdot n \cdot n$. Below are the 2 Characteristics.

For any integer number N,

  1. Last digit of $N$ and its last digit of fifth power $N \cdot N \cdot N \cdot N \cdot N$ are same.
  2. Value of $(N\cdot N\cdot N\cdot N \cdot N) - N$ is always divisible by $30$.

Few Examples below,

  • $N = 2$,

    $$(N\cdot N\cdot N\cdot N \cdot N) = 32$$

    $$\left(\frac{(N\cdot N\cdot N\cdot N \cdot N)-N}{30}\right) = 1$$

  • $N = 4$

    $$(N\cdot N\cdot N\cdot N \cdot N) = 1024$$

    $$\left(\frac{(N\cdot N\cdot N\cdot N \cdot N)-N)}{30}\right) = 34$$

If this findings are not valid please defend this statement with your examples.

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What is ghandhan? – Myself Apr 19 '13 at 9:11

These statements are true but almost trivial to prove. Nothing significant or earth-shattering here.

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