What is the Remainder of 2 to the Power of 1000005 Divided by 55?

[zetafunction]

Homework Statement

i need to obtain the remainder of the divison [tex] 2^{1000005}[/tex] divided by 55

Homework Equations

Euler theorem [tex] 2^{\phi (55)}=1 mod(55) [/tex]

The Attempt at a Solution

my problem is that applying Euler theorem i reach to the conclusion that the remainder is the same as the value 'a' inside the congruence equation

[tex] 2^{5}=a mod(55)[/tex] but it would give me that a is negative ¡¡

it gives me a=-23 or similar using congruences or a =32

 

[Count Iblis]

phi(55) = 4*10=40

So, 2^40 = 1

1000005 = 1000000 + 5 which Modulo 40 is 5

So 2^2000005 = 2^5 = 32.