To convert the base ten numeral $729$ to base seven, we can use the method of successive division by 7, keeping track of the remainders. Here's how you can do it:

1. Divide 729 by 7: $729 \div 7 = 104 \quad \text{with a remainder of } 1$ (Write down the remainder $1$).

2. Divide the quotient 104 by 7: $104 \div 7 = 14 \quad \text{with a remainder of } 6$ (Write down the remainder $6$).

3. Divide the quotient 14 by 7: $14 \div 7 = 2 \quad \text{with a remainder of } 0$ (Write down the remainder $0$).

4. Divide the quotient 2 by 7: $2 \div 7 = 0 \quad \text{with a remainder of } 2$ (Write down the remainder $2$).

Since the quotient is now $0$, we stop the division process. The base seven numeral is obtained by reading the remainders from the last to the first. Therefore, $729$ in base ten is equal to $2061_7$ in base seven.

Summary:

• $729 \div 7 = 104$, remainder $1$
• $104 \div 7 = 14$, remainder $6$
• $14 \div 7 = 2$, remainder $0$
• $2 \div 7 = 0$, remainder $2$

Final Answer: $729_{10} = 2061_7$

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Would you like more details or have any questions?

Here are some related questions you might find useful:

1. How do you convert the base ten numeral 100 to base two?
2. What is the base eight equivalent of the decimal number 512?
3. How do you convert a base seven numeral back to a base ten numeral?
4. Explain the process of converting base ten to base five.
5. What is the base three equivalent of the decimal number 243?
6. Can you convert the decimal number 81 to base nine?
7. How would you convert the number 625 from base ten to base four?
8. Why is the base seven numeral system useful in some calculations?

Tip: When converting from base ten to any other base, always read the remainders from bottom to top to get the final numeral in the new base.