PLease help! Use Fermat's or Euler's theorem to calculate each of the following.?

Question

I need to explain which theorem I'm using and mhy I didn't use the other!
2^5 (mod 5)
3^22 (mod23)
3^65537 (mod 65537) 65537 is prime
7^55 (mod 19)

Extremely grateful for any help. Thanks

Answer 1

if p is a pime

a^p = a mod p

so 2^5 = 2 mod 5

and 3^65537 = 3 mod 65537

additionally

a^p-1 = 1 mod p

so 3^22 = 1 mod 23 as 23 is prime

now 7^55 mod 19

7^18 mod 19 = 1
cube both sides
7^54 mod 19 = 1
multiply by 7
7^55 mod 19 = 7

Answer 2

Since all these moduli are prime, we can use
Fermat's little theorem for all of these:
So
2^5 = 2(mod 5)
3^22 = 1(mod 23)
3^65537 = 3(mod 65537)
The last one is the only one requiring a bit of work.
We know 7^18 = 1(mod 19) by Fermat's theorem.
So 7^54 = (7^18)^3 = 1(mod 19)
Thus 7^55 = 7(mod 19)

Answer 3

Fermat's theorem applies to when the exponent is a prime number. Euler's theorem applies to a more general case. For more info see...
http://en.wikipedia.org/wiki/Euler's_theorem
While you're there, also click on the link for Fermat's Little Theorem for comparison.

Answer 4

SO glad i'm not in that class! LOL! no offense.

http://answers.yahoo.com/question/index;_ylt=AvrbEp0KW9gPuppEZ.DdAXnsy6IX?qid=20070514201649AAz9wuH

Answer 5

Get ******!!!!