To convert the base-10 number $34563$ to base-6, follow these steps:

Step 1: Divide by 6 and Record Remainders

We repeatedly divide the number by 6 and keep track of the remainders:

$34563 \div 6 = 5760 \text{ remainder } 3$ $5760 \div 6 = 960 \text{ remainder } 0$ $960 \div 6 = 160 \text{ remainder } 0$ $160 \div 6 = 26 \text{ remainder } 4$ $26 \div 6 = 4 \text{ remainder } 2$ $4 \div 6 = 0 \text{ remainder } 4$

Step 2: Read the Remainders from Bottom to Top

The base-6 representation is:

$\mathbf{424003_6}$

Thus, the base-10 number 34563 is 424003 in base-6.

Would you like a breakdown of why this method works?

Related Questions:

1. How do you convert numbers from base-6 back to base-10?
2. What is the base-6 representation of $50000$?
3. How do place values work in different bases?
4. Can this method be applied to non-integer numbers?
5. How do you perform arithmetic (addition, subtraction) in base-6?

Tip:

When converting a number to a different base, always start from the largest place value and work downward!