The last digit of 17 to the 17th

Idea. The last digit of a product depends only on the last digits of the factors, and the last digits of the powers of 7 repeat with period 4.

Step 1. $17^{17}$ ends with the same digit as $7^{17}$.

Step 2. The powers of 7 end in $7,9,3,1$, and then the cycle repeats.

Step 3. $17=4\times 4+1$, so $7^{17}$ ends like $7^1$: the last digit is 7.

In general. The last digits of powers cycle with period at most 4, so look at the exponent modulo 4. For example, $3^{33}$ ends in 3, since $33\equiv 1(\mathrm{mod}4)$.