Determine the rightmost decimal digit of \[ 17^{17^{17}}. \]
Solution
Set \[ N\coloneqq 17^{17}. \]
1. One has\[ N\equiv1\pmod4. \]
Since \[ 17\equiv1\pmod4, \] raising both sides to the seventeenth power gives \[ 17^{17}\equiv1^{17}\equiv1\pmod4. \]
2. One has\[ 17^N\equiv7\pmod{10}. \]
Modulo \(10\), \[ 17^N\equiv7^N. \] Also \[ 7^4=2401\equiv1\pmod{10}. \] By step 1, there is an integer \(q\) such that \(N=4q+1\). Therefore \[ 7^N = 7(7^4)^q \equiv 7\pmod{10}. \]
3. The rightmost decimal digit of \(17^{17^{17}}\) is\[ \boxed{7}. \]
The number in the problem is \(17^N\), and step 2 says that it is congruent to \(7\) modulo \(10\).
4. Q.E.D.
Step 3 gives the requested digit.