# How do you find the last digit of a number using Fermat's Little Theorem

• Dec 11th 2009, 02:45 PM
steph3824
How do you find the last digit of a number using Fermat's Little Theorem
For example, what is the last digit of 17^(3241)? It would help a ton if someone could show how to solve this using Fermat's Little Theorem. I have finals coming up and this was something I never quite understood
• Dec 12th 2009, 04:18 AM
CaptainBlack
Quote:

Originally Posted by steph3824
For example, what is the last digit of 17^(3241)? It would help a ton if someone could show how to solve this using Fermat's Little Theorem. I have finals coming up and this was something I never quite understood

I'm not sure why one would need Fermat's little theorem for this since:

$17^3 \equiv 1 \text{ mod } 10$

and:

$17^{3241}=(17^3)^{1080}\times 17$

which immediately should tell you that:

$17^{3241}\equiv 7 \text{ mod } 10$

CB
• Dec 12th 2009, 05:40 AM
alexmahone
Quote:

Originally Posted by CaptainBlack
I'm not sure why one would need Fermat's little theorem for this since:

$17^3 \equiv 1 \text{ mod } 10$

and:

$17^{3241}=(17^3)^{1080}\times 17$

which immediately should tell you that:

$17^{3241}\equiv 7 \text{ mod } 10$

CB

Incorrect!

$17^4 \equiv 1 (mod 10)$

$17^{3241}=(17^4)^{810}\times 17$

So $17^{3241}\equiv 7(mod 10)$
• Dec 12th 2009, 12:24 PM
CaptainBlack
Quote:

Originally Posted by alexmahone
Incorrect!

$17^4 \equiv 1 (mod 10)$

$17^{3241}=(17^4)^{810}\times 17$

So $17^{3241}\equiv 7(mod 10)$

Oppsss.. arithmetic error (actually a transcription error with argument completed with the erroeous value)

CB
• Dec 13th 2009, 06:52 AM
awkward
Quote:

Originally Posted by steph3824
For example, what is the last digit of 17^(3241)? It would help a ton if someone could show how to solve this using Fermat's Little Theorem. I have finals coming up and this was something I never quite understood

Here is a solution via Fermat's Last Theorem (although it's not necessarily the simplest):

By Fermat's Last Theorem, $17^4 \equiv 1 \mod 5$,
so since
$17^{3241} = (17^4)^{810} \cdot 17$,
$17^{3241} \equiv 1 \cdot 17 \equiv 2 \mod 5$.

We also have $17 \equiv 1 \mod 2$, so $17 ^ {3241} \equiv 1 \mod 2$.

Since $17^{3241} \equiv 2 \mod 5$ and $17 ^ {3241} \equiv 1 \mod 2$,
$17^{3241} \equiv 7 \mod 10$.
We could use the Chinese Remainder Theorem at the last step, but it seems like overkill here.