To convert $27$ from base $10$ to base $2$:

Steps:

1. Divide $27$ by $2$ and record the quotient and remainder: $27 \div 2 = 13 \text{ remainder } 1$
2. Divide the quotient $13$ by $2$: $13 \div 2 = 6 \text{ remainder } 1$
3. Divide $6$ by $2$: $6 \div 2 = 3 \text{ remainder } 0$
4. Divide $3$ by $2$: $3 \div 2 = 1 \text{ remainder } 1$
5. Divide $1$ by $2$: $1 \div 2 = 0 \text{ remainder } 1$

Write the remainders from bottom to top:

The binary representation of $27$ is: $27_{10} = 11011_2$

Final Answer:

$11011_2$

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Would you like a more detailed breakdown or have any questions?

Here are 5 related questions for further exploration:

1. How do you convert a binary number back to decimal?
2. How would $27$ look in base $16$ (hexadecimal)?
3. Can you explain how binary addition works with $27_2 + 5_2$?
4. What is the binary representation of $27$ divided by $3$?
5. How is binary used in computer systems?

Tip: When converting to base $2$, the remainders represent the digits of the binary number in reverse order from the division process!