Finding the least significant digit of a large exponential. I am trying to find the least significant digit of $17^{{17}^{17}}$. I know that I need to use to use the properties of modular arithmetic and mod base 10, but I am not sure how to go about it. 

Please provide some hints/first few steps to help me get started.
Start off by looking at $17^n$ mod 10. (In your case, n will end up being $17^{17}$, but that's way too big to calculate yet.)

$17^0$ ends in a $1$, $17^1$ ends in a $7$, $17^2$ is congruent to $7 \times 7$ so it ends in a $9$, $17^3$ likewise is congruent to $9 \times 7$ so it ends in a $3$, and finally $17^4$ is congruent to $3 \times 7$ so it ends in a $1$.

Since $17^0$ and $17^4$ are congruent mod 10, it follows that $17^n$ mod 10 will repeat every time the exponent $n$ goes up by 4.

Therefore, to solve your problem, you now need to calculate the exponent $17^{17}$ mod 4. Then you can use that along with the pattern I just described to get the final answer. Since this is homework, I'll let you calculate $17^{17}$ mod 4 yourself... hint, use the same idea that I used above!
Hint: For distinct primes $p, q,$ and any $a \neq 0
(\mod pq)$, use the formula: 
$$a^{(p-1)}(q-1) = 1 (\mod pq)$$