# A problem with Euler theorem

## Homework Statement

i need to obtain the remainder of the divison $$2^{1000005}$$ divided by 55

## Homework Equations

Euler theorem $$2^{\phi (55)}=1 mod(55)$$

## 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

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

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

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phi(55) = 4*10=40

So, 2^40 = 1

1000005 = 1000000 + 5 which Modulo 40 is 5

So 2^2000005 = 2^5 = 32.