If 2 numbers a=b mod(n), does that imply they are congruent modulo as well?

Anyway on to the matter at hand. Calculate 2^258 (mod 259)I did it 2 different ways and unfortunatly got 2 different answers.

2^(64)=86 mod 259,

2^(128)=144,

2^(256)=16 so 2^256.2^2=64 (mod 259). I know my method isn't very clear but you probably get the idea.

Method 2

259 = 7 x 37

2^6=1 mod(7)

2^258=(2^6)^43=1 mod (7)

2^36=1 mod(37)

2^258=(2^36)^7.2^6=27 mod(37)

but 1 x 27=27 not 64.

I have a strong feeling my first method is correct. Which one is correct and why is the other wrong?