To convert the decimal number $200,000_{10}$ to base 13, follow these steps:

1. Divide 200,000 by 13 and note the quotient and remainder. $200,000 \div 13 = 15,384 \text{ remainder } 8$ So, the least significant digit is $8$.

2. Divide the quotient (15,384) by 13 and continue noting the remainders: $15,384 \div 13 = 1,183 \text{ remainder } 5$ So, the next digit is $5$.

3. Divide the quotient (1,183) by 13: $1,183 \div 13 = 91 \text{ remainder } 0$ The next digit is $0$.

4. Divide the quotient (91) by 13: $91 \div 13 = 7 \text{ remainder } 0$ The next digit is $0$.

5. Finally, divide 7 by 13: $7 \div 13 = 0 \text{ remainder } 7$ So, the most significant digit is $7$.

Therefore, the base-13 equivalent of $200,000_{10}$ is $70,058_{13}$.

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Related Questions:

1. How would you convert a number from base 13 back to base 10?
2. What are other efficient methods for base conversion?
3. How does base conversion work for fractional numbers?
4. Why do some systems use different number bases (e.g., base-13)?
5. How can you generalize this process for any base conversion?

Tip:

When converting large numbers, it's useful to use a calculator or software for efficient computation, especially for non-standard bases like base-13.