# Final Digit of 3^1001

• May 24th 2008, 11:37 AM
gearshifter
Final Digit of 3^1001
Explaining your logic, what is the final digit of the number 3^1001?

*I'm guessing that they don't want you to use a calculator?

Thank you.
• May 24th 2008, 11:47 AM
Moo
Hello,

Quote:

Originally Posted by gearshifter
Explaining your logic, what is the final digit of the number 3^1001?

*I'm guessing that they don't want you to use a calculator?

Thank you.

You can see it by iterations...

$\displaystyle 3^1=\bf{3}$
$\displaystyle 3^2=\bf{9}$
$\displaystyle 3^3=2\bf{7}$
$\displaystyle 3^4=8\bf{1}$
$\displaystyle 3^5=24\bf{3}$
$\displaystyle 3^6=72\bf{9}$

So you can see that there are only 4 possible values : 1;3;7;9. When you multiply by 3 a number which finishes by 3, the result will finish by 9. When you multiply by 3 a number which finishes by 9, the result will finish by 7, etc... You can prove it by writing that these numbers are $\displaystyle 10k+3$ for example.. When multiplying by 3, it yields $\displaystyle 30k+9$, which obviously has its final digit number equal to 9 (Wink)
You can notice that the same value comes back every time you add 4 to the power.

For example : $\displaystyle 3^1, 3^5, 3^9, \dots$ will have the same final digit number.

So continuing this way, if I add 25 times 4 to the power, it will be $\displaystyle 3^{101}$, which will have the same final digit number as $\displaystyle 3^1$

What is it going to be for $\displaystyle 3^{1001}$ ?

(I hope this is clear enough :/)