Find the fifth digit from the end of the number below.

$\Large { 5 }^{ { 5 }^{ { 5 }^{ { 5 }^{ 5 } } } }$

The answer is 0.

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My approach is similar to Priyanshu's, but I will organize my work a bit differently.

Clearly, the given number $N$ is divisible by $5^5=3125$ .

To find the congruency class of $N$ modulo $2^5=3$ 2, we can work module $\phi(32)=16$ in the first exponent, modulo $\phi(16)=8$ in the second exponent, modulo 4 in the fourth exponent, and modulo 2 in the top exponent. Thus we have $N\equiv 5^{5^5}\equiv 5^5 \pmod {32}$

Since $N\equiv 5^5$ both modulo $5^5$ and modulo $2^5$ , the congruency holds modulo $2^5\times 5^5=10^5=10000$ as well, so that $N$ ends in $...03125$ , and the fifth digit from the end is $\boxed{0}$