In a base-4 system, what is the sum of the numbers '12' and '21'?

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Q: In a base-4 system, what is the sum of the numbers '12' and '21'?

Solution: The sum of '12' and '21' in base-4 is '30' in base-4.

Steps: 9

Step 1: Understand that we are working in base-4. This means the digits can only be 0, 1, 2, or 3.
Step 2: Write down the numbers we want to add: '12' (base-4) and '21' (base-4).
Step 3: Convert '12' from base-4 to base-10. '12' in base-4 means (1 * 4^1) + (2 * 4^0) = 4 + 2 = 6 in base-10.
Step 4: Convert '21' from base-4 to base-10. '21' in base-4 means (2 * 4^1) + (1 * 4^0) = 8 + 1 = 9 in base-10.
Step 5: Add the two base-10 numbers together: 6 + 9 = 15 in base-10.
Step 6: Convert the sum (15 in base-10) back to base-4. To do this, divide 15 by 4. The quotient is 3 and the remainder is 3. So, we write down 3.
Step 7: Now take the quotient (3) and divide it by 4. The quotient is 0 and the remainder is 3. So, we write down 3 again.
Step 8: Read the remainders from last to first. The base-4 representation of 15 is '33'.
Step 9: Therefore, the sum of '12' and '21' in base-4 is '33'.