Number Theory

Let's write out the first few terms and see if we can observe a pattern with the last three digits.

$5^{1} mod 1000 =5$

$5^{2} mod 1000 =25$

$5^{3} mod 1000 =125$

$5^{4} mod 1000 =625$

$5^{5} mod 1000 =125$

$5^{6} mod 1000 =625$

We observe that the last three digits for every odd value of n is 125 while the last three digits for every even value of n is 625. We then need to determine the parity of the initial value (Whether it is even or odd)

We know that any odd integer multiplied by any odd integer is going to have an odd parity, so $5^{5^{5^{5}}}$ is going to end up being odd because it is simply 5 multiplied by itself $5^{5^{5}}$ times. Therefore, we know that the last three digits must be 125 .