In a base-5 number system, what is the sum of '34' and '21'?

In a base-5 number system, what is the sum of '34' and '21'?

Step 1: Understand that we are working in base-5. This means each digit represents a power of 5.
Step 2: Convert '34' from base-5 to decimal. Calculate 3*5^1 + 4*5^0 = 3*5 + 4*1 = 15 + 4 = 19.
Step 3: Convert '21' from base-5 to decimal. Calculate 2*5^1 + 1*5^0 = 2*5 + 1*1 = 10 + 1 = 11.
Step 4: Add the two decimal numbers together: 19 + 11 = 30.
Step 5: Convert the sum (30) back to base-5. Find the largest power of 5 less than or equal to 30, which is 25 (5^2).
Step 6: Determine how many times 25 fits into 30. It fits 1 time, so we write down '1' for the 5^2 place.
Step 7: Subtract 25 from 30, which gives us 5. Now, find how many times 5 (5^1) fits into 5. It fits 1 time, so we write down '1' for the 5^1 place.
Step 8: Subtract 5 from 5, which gives us 0. Now, we have no remainder, so we write down '0' for the 5^0 place.
Step 9: Combine the digits we found: '100' in base-5.

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