Find the last digit of $66^5$

Question

Find the last digit of $66^5$.

This is how I solved the problem: $66^5=6^5*11^5$ (mod 10) = $6^5*1^5$ (mod 10). I have two questions. First, what is wrong with my method? I get different answer from answer sheet. Second, is there an easy way to find the last digit of $6^5$?

Hint $\ $ By induction $\ a^2\equiv a,\Rightarrow, a^n \equiv a\ $ for all $,n \ge 1\ \ $ — Bill Dubuque, Jun 18 '16 at 15:26
Or: $\ 6^n\equiv 0\pmod 2,$ and $,6^n \equiv 1^n\equiv\color{#c00} 1\pmod{5},$ so $,6^n \equiv \color{#c00}1+5\pmod{10}\ \ $ — Bill Dubuque, Jun 18 '16 at 15:31

Roby5 · Answer 1 · 2016-06-18T12:05:37.640

3

Hint:

Observe that $6^n \equiv 6 \pmod {10}$, where $n$ is a positive integer.

With this hint, it may be seen that nothing is wrong with your answer as

$66^5 =6^5 \cdot 11^5 \equiv 6^5 \cdot 1 \equiv 6 \pmod {10}$

edited Jun 18 '16 at 12:05

answered Jun 18 '16 at 11:52

Roby5

4,287

@learning Yes, $6^2=36 \equiv 6\pmod {10}$ – Roby5 Jun 18 '16 at 11:58

score 2 · Answer 2 · edited Apr 13 '17 at 12:20

2

For integer $n\ge0,$ $$6^{n+1}-6=6(6^n-1^n)\equiv0\pmod{10}$$ using Why $a^n - b^n$ is divisible by $a-b$?

edited Apr 13 '17 at 12:20

Community

1

answered Jun 18 '16 at 11:52

lab bhattacharjee

274,582

Find the last digit of $66^5$

2 Answers2