# Thread: Fermat’s Little Theorem Problem

1. ## Fermat’s Little Theorem Problem

Using Fermat’s Little Theorem, or, otherwise, find the remainder when 3^203 is divided by 11. Thanks for your help in advance.

2. Originally Posted by MathStudent1
Using Fermat’s Little Theorem, or, otherwise, find the remainder when 3^203 is divided by 11. Thanks for your help in advance.
11 does not divide 3.

Thus,

3^10 = 1 (mod 11)

Thus,

(3^10)^20 = 1^20 (mod 11)

Thus

3^200 = 1 (mod 11)

Thus,

3^203 = 3^3 = 27 = 5 (mod 11)

3. TPH,

Can you tell me how you went from 27 in the last line and then you got 5 from there? I understand the mod(11) part. Thanks!

4. Originally Posted by MathStudent1
TPH,

Can you tell me how you went from 3^203 to 3^3 in the last line? Then I know 3^3 is 27 and then how did you get 5 from there? I understand the mod(11) part. Thanks!
Okay, = is supposed to mean 'congruent to' instead of 'equals'.

I multiplied both sides by 3^3. And then 3^3=27 and under mod 11 that is 5.

5. Got it! Thanks.