How can I use eulers totient and the chinese remainder theorem for modular exponentiation?
To step through it simply:

Step 1: we first compute compute $M^e \bmod p$ (where $p$ is one of the factors of the modulus.  By Fermat's Little Theorem, this is the same as $M^{e \mod p-1} \bmod p$ (and that's where the real savings are), and so in your example ($p=5$), we get:


$M^e \bmod p = 27^{37 \bmod 4} \bmod 5 = 2^1 \bmod 5 = 2$


Step 2: we do the same for the other prime $q$:


$M^e \bmod q = 27^{37 \bmod 10} \bmod 11 = 5^7 \bmod 11 = 3$


Step 3: using the Chinese Remainder theorem, we reconstruct $M^e \bmod pq$ from $M^e \bmod p$ and $M^e \bmod q$:


$C_p = M^e \bmod p = 2$
$C_q = M^e \bmod q = 3$
$M^e \bmod pq = C_q + q \cdot (q^{-1} (C_p - C_q) \bmod p) = 3 + 11 \cdot (1 \cdot (2-3) \bmod 5) = 3 + 11 \cdot 4 = 47$