What is the base 4 representation of the base 2 number $11011000_2$?
Instead of converting to base 10 and then to base 4, we use the fact that $2^2=4$. We have $11011000_2=1\cdot2^7+1\cdot2^6+1\cdot2^4+1\cdot2^3$ $=2\cdot(2^2)^3+1\cdot(2^2)^3+1\cdot(2^2)^2+2\cdot(2^2)^1$ $=3\cdot4^3+1\cdot4^2+2\cdot4^1+0\cdot4^0=\boxed{3120_4}$.
3120_4
Level 3
Number Theory
Problem: What is the base 4 representation of the base 2 number $11011000_2$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To convert the base 2 number $11011000_2$ to base 4, first recognize that base 4 is a power of base 2, specifically $4 = 2^2$. This means that each base 4 digit corresponds to two base 2 digits. 

Start by grouping the base 2 number into pairs from the right: $11\ 01\ 10\ 00$. If there is an odd number of digits, pad the leftmost side with a zero. In this case, we have four pairs: $11$, $01$, $10$, and $00$.

Next, convert each pair of base 2 digits to base 4:
- The pair $11_2$ converts to $3_4$ (since $1 \cdot 2^1 + 1 \cdot 2^0 = 2 + 1 = 3$).
- The pair $01_2$ converts to $1_4$ (since $0 \cdot 2^1 + 1 \cdot 2^0 = 0 + 1 = 1$).
- The pair $10_2$ converts to $2_4$ (since $1 \cdot 2^1 + 0 \cdot 2^0 = 2 + 0 = 2$).
- The pair $00_2$ converts to $0_4$ (since $0 \cdot 2^1 + 0 \cdot 2^0 = 0 + 0 = 0$).

Now, combine the base 4 digits obtained from the conversions in the same order as the pairs were grouped: $3$, $1$, $2$, and $0$. 

Thus, the base 4 representation of the base 2 number $11011000_2$ is $3120_4$.<|END_OF_LATENT|> The final answer is $3120_4$.