Find the last two digits of:

$\large{25^{\underbrace{252525 \ldots 25}_{50 \text{ digits}}}}$

The answer is 25.

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From $25^1 = 25$ , $25^2 = 625$ , and $25^3 = 15625$ , it would appear that the last two digits $25^n$ are $25$ for any positive integer $n$ .

This can be proved inductively:

Therefore, the last two digits of the given number are $\boxed{25}$ .

Bonus:$25$ is another automorphic number .