The problem is asking to convert the base-10 number $1642$ into a base-7 number. Let's walk through the process of conversion.

Steps to Convert from Base-10 to Base-7:

1. Divide the number by 7 and record the quotient and remainder.
2. Repeat the division process on the quotient until the quotient is 0.
3. The base-7 number will be the sequence of remainders read from bottom to top.

Calculation:

1. $1642 \div 7 = 234$ remainder $4$
2. $234 \div 7 = 33$ remainder $3$
3. $33 \div 7 = 4$ remainder $5$
4. $4 \div 7 = 0$ remainder $4$

Now, reading the remainders from bottom to top, we get the base-7 number:

Result: $4534_7$

So, $1642_{10} = 4534_7$.

Would you like a further breakdown of the steps? Let me know if you need details or have any questions!

Here are five related questions:

1. How would you convert a base-7 number back to base-10?
2. What is the base-7 equivalent of $2000$?
3. How do you convert a base-10 fraction to base-7?
4. What is the difference between base-7 and base-8 number systems?
5. How can you check if a conversion to base-7 is correct?

Tip: Always verify your result by converting the base-7 number back to base-10 for accuracy!