Okay, so I need to divide two numbers in base 5 and express the quotient also in base 5. The numbers given are 413 base 5 and 2 base 5. Hmm, base 5, right? Let me recall how division works in different bases. I think the process is similar to long division in base 10, but I have to remember that each digit represents powers of 5 instead of 10. Let me start by converting both numbers to base 10 to make the division easier, then convert the result back to base 5.

First, let's convert 413_5 to base 10. The rightmost digit is the units place (5^0), the middle digit is the fives place (5^1), and the leftmost digit is the twenty-fives place (5^2). So, breaking it down:

4 * 5^2 + 1 * 5^1 + 3 * 5^0
= 4 * 25 + 1 * 5 + 3 * 1
= 100 + 5 + 3
= 108 in base 10.

Next, convert 2_5 to base 10. That's straightforward since it's a single digit. 2 * 5^0 = 2 * 1 = 2.

Now, divide 108 by 2 in base 10. 108 ÷ 2 = 54. So the quotient is 54 in base 10. Now, I need to convert 54 back to base 5.

To convert a base 10 number to base 5, I divide the number by 5 and keep track of the remainders. Let's do that step by step.

54 ÷ 5 = 10 with a remainder of 4.
10 ÷ 5 = 2 with a remainder of 0.
2 ÷ 5 = 0 with a remainder of 2.

Now, writing the remainders from last to first, we get 2 0 4. So 54 in base 10 is 204_5.

Wait, let me verify that. Let's check 204_5 in base 10:

2 * 5^2 + 0 * 5^1 + 4 * 5^0
= 2 * 25 + 0 * 5 + 4 * 1
= 50 + 0 + 4
= 54. Yes, that's correct.

But just to make sure, maybe I should try performing the division directly in base 5 without converting to base 10. Sometimes, direct division can help confirm the result. Let's attempt that.

Dividing 413_5 by 2_5 in base 5.

First, let's write the numbers:

      ___
2_5 |413_5

Starting from the leftmost digit of the dividend. The first digit is 4. Since we're in base 5, 2 goes into 4 how many times? In base 10, 2*2=4, so 2 times. But in base 5, 2 times 2 is 4, which is 4_5. So we write 2 as the first digit of the quotient.

Multiply 2 (quotient digit) by 2 (divisor) to get 4. Subtract that from 4, which gives 0. Bring down the next digit, which is 1. Now we have 01 in base 5.

But wait, in base 5, 01 is just 1. How many times does 2 go into 1? It doesn't. So we write 0 as the next digit in the quotient. But wait, in base 5 division, if the divisor is larger than the current dividend, we write 0 and bring down the next digit. So bring down the 3, making it 13_5.

Now, how many times does 2 go into 13_5? Let's convert 13_5 to base 10: 1*5 + 3 = 8. 2 goes into 8 four times. So 4 times. But in base 5, 4 is 4_5. So multiply 2_5 by 4_5. Let's compute that in base 5.

2_5 * 4_5. In base 10, 2*4=8. 8 divided by 5 is 1 with remainder 3, so that's 1*5 + 3 = 13_5. So the product is 13_5. Subtract that from 13_5 (the current dividend), which gives 0. So the quotient is 204_5 with no remainder. That matches the previous result. So the answer is 204_5.

Therefore, both methods give the same answer. So I can be confident that 413_5 ÷ 2_5 is 204_5. Let me just recap to make sure I didn't make any mistakes in the direct division.

First step: 2 into 4 in base 5 is 2, subtract 4, remainder 0. Bring down 1. 2 into 1 in base 5 is 0, remainder 1. Bring down 3. 2 into 13 (base 5, which is 8 in base 10) is 4 (since 2*4=8). Multiply 2*4=13 in base 5. Subtract 13 from 13, remainder 0. So quotient digits are 2, 0, 4. So 204_5. All steps check out. So the answer is correct.

Alternatively, maybe I can check by multiplying 2_5 by 204_5 and see if I get 413_5.

Multiplying 2_5 by 204_5:

First, convert 204_5 to base 10: 2*25 + 0*5 + 4 = 54.

Multiply 54 by 2 in base 10: 108.

Convert 108 back to base 5: 108 ÷ 25 = 4 with remainder 8; 8 ÷ 5 = 1 with remainder 3; 3 ÷ 1 = 3. So 4*25 + 1*5 + 3*1 = 413_5. Which matches the original dividend. So that's correct.

Everything checks out. So the final answer is 204_5.
Thus, the quotient of $413_5 \div 2_5$ expressed in base 5 is $\boxed{204_5}$.
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