<think>
Okay, so I need to divide 413 base 5 by 2 base 5 and give the answer in base 5. Hmm, let me think. First, maybe I should convert both numbers to base 10, do the division there, and then convert the result back to base 5? That might be easier for me since I'm more comfortable with base 10 arithmetic. Let me try that approach.

Starting with the dividend, which is 413 in base 5. To convert that to base 10, I need to remember that each digit represents a power of 5. The rightmost digit is 5^0, then 5^1, then 5^2, and so on. So let's break down 413_5:

The digits are 4, 1, 3 from left to right. So that would be:

4 * 5^2 + 1 * 5^1 + 3 * 5^0.

Calculating each term:

First term: 4 * 5^2. 5 squared is 25, so 4 * 25 = 100.

Second term: 1 * 5^1 = 1 * 5 = 5.

Third term: 3 * 5^0 = 3 * 1 = 3.

Adding them up: 100 + 5 + 3 = 108. So 413 base 5 is 108 in base 10.

Now the divisor is 2_5. Converting that to base 10 is straightforward: 2 * 5^0 = 2 * 1 = 2. So 2 base 5 is 2 in base 10.

So now the problem becomes 108 divided by 2 in base 10. Let me do that division. 108 divided by 2 is 54. So the result in base 10 is 54. Now I need to convert 54 back to base 5.

To convert 54 to base 5, I need to find the coefficients for the powers of 5. Let me recall that 5^0 = 1, 5^1 = 5, 5^2 = 25, 5^3 = 125. Since 5^3 is 125 which is larger than 54, the highest power needed is 5^2 = 25.

So let's divide 54 by 25 to find the coefficient for 5^2. 54 divided by 25 is 2 with a remainder. 2*25 = 50. So the coefficient for 5^2 is 2. Then subtract 50 from 54, which leaves 4.

Now, moving to the next lower power, which is 5^1 = 5. Take the remainder 4 and divide by 5. That's 0 with a remainder of 4. So the coefficient for 5^1 is 0.

Then the remainder is 4, which is the coefficient for 5^0. So putting it all together, the base 5 number is 2 (for 5^2), 0 (for 5^1), and 4 (for 5^0). Therefore, 54 in base 10 is 204 in base 5.

Wait, let me check that again. Let me verify the conversion. 2*25 + 0*5 + 4*1 = 50 + 0 + 4 = 54. Yes, that's correct. So 54 base 10 is 204 base 5. Therefore, the quotient of 413_5 divided by 2_5 is 204_5.

But hold on, maybe I should try doing the division directly in base 5 to confirm? Sometimes converting back and forth can introduce errors, especially if I made a mistake in conversion. Let me try that method as well to verify.

So, let's perform the division 413_5 ÷ 2_5 directly in base 5. Let me recall how division works in different bases. It's similar to base 10, but all the operations are done in base 5.

First, let me write down the dividend and divisor:

Dividend: 413_5

Divisor: 2_5

So, we need to divide 413_5 by 2_5. Let me set it up like long division.

First, I need to see how many times 2_5 goes into the first digit of the dividend. The first digit is 4_5. Let me convert 4_5 to base 10 to understand its value. 4 in base 5 is 4 in base 10. So 4 divided by 2 is 2. So 2_5 goes into 4_5 twice. So I write 2 as the first digit of the quotient.

Multiply 2_5 by 2_5. Let me compute that in base 5. 2*2 is 4, so 4_5. Subtract that from 4_5, which gives 0. Bring down the next digit, which is 1_5. So now we have 01_5.

Now, how many times does 2_5 go into 1_5? Well, 2_5 is 2 in base 10, and 1_5 is 1 in base 10. So 2 goes into 1 zero times. So we write 0 as the next digit in the quotient. Multiply 0 by 2_5, which is 0, subtract that from 1_5, giving 1_5. Bring down the next digit, which is 3_5. So now we have 13_5.

Now, how many times does 2_5 go into 13_5? Let me convert 13_5 to base 10 to make it easier. 1*5 + 3 = 8. And 2_5 is 2 in base 10. So 8 divided by 2 is 4. So 4 times. So we write 4 as the next digit in the quotient. Multiply 4_5 by 2_5. Let me compute that in base 5. 4*2 is 8 in base 10. Now, convert 8 to base 5. 5 goes into 8 once with a remainder of 3. So 8 is 13_5. Therefore, 4_5 * 2_5 = 13_5. Subtract that from 13_5, which gives 0. So there's no remainder.

Therefore, the quotient is 2 0 4_5, which is 204_5. That matches the result I got earlier when converting to base 10. So that seems to confirm that the answer is indeed 204_5.

But just to be thorough, let me check once more by converting 204_5 back to base 10 and see if it equals 54. So 2*5^2 + 0*5^1 + 4*5^0 = 2*25 + 0 + 4 = 50 + 4 = 54. Which is correct. And 54 divided by 2 is 27? Wait, no, wait. Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? 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[Assistant]
<think>
Okay, so I need to divide 413 base 5 by 2 base 5 and give the answer in base 5. Hmm, let me think. First, maybe I should convert both numbers to base 10, do the division there, and then convert the result back to base 5? That might be easier for me since I'm more comfortable with base 10 arithmetic. Let me try that approach.

Starting with the dividend, which is 413 in base 5. To convert that to base 10, I need to remember that each digit represents a power of 5. The rightmost digit is 5^0, then 5^1, then 5^2, and so on. So let's break down 413_5:

The digits are 4, 1, 3 from left to right. So that would be:

4 * 5^2 + 1 * 5^1 + 3 * 5^0.

Calculating each term:

First term: 4 * 5^2. 5 squared is 25, so 4 * 25 = 100.

Second term: 1 * 5^1 = 1 * 5 = 5.

Third term: 3 * 5^0 = 3 * 1 = 3.

Adding them up: 100 + 5 + 3 = 108. So 413 base 5 is 108 in base 10.

Now the divisor is 2_5. Converting that to base 10 is straightforward: 2 * 5^0 = 2 * 1 = 2. So 2 base 5 is 2 in base 10.

So now the problem becomes 108 divided by 2 in base 10. Let me do that division. 108 divided by 2 is 54. So the result in base 10 is 54. Now I need to convert 54 back to base 5.

To convert 54 to base 5, I need to find the coefficients for the powers of 5. Let me recall that 5^0 = 1, 5^1 = 5, 5^2 = 25, 5^3 = 125. Since 5^3 is 125 which is larger than 54, the highest power needed is 5^2 = 25.

So let's divide 54 by 25 to find the coefficient for 5^2. 54 divided by 25 is 2 with a remainder. 2*25 = 50. So the coefficient for 5^2 is 2. Then subtract 50 from 54, which leaves 4.

Now, moving to the next lower power, which is 5^1 = 5. Take the remainder 4 and divide by 5. That's 0 with a remainder of 4. So the coefficient for 5^1 is 0.

Then the remainder is 4, which is the coefficient for 5^0. So putting it all together, the base 5 number is 2 (for 5^2), 0 (for 5^1), and 4 (for 5^0). Therefore, 54 in base 10 is 204 in base 5.

Wait, let me check that again. Let me verify the conversion. 2*25 + 0*5 + 4*1 = 50 + 0 + 4 = 54. Yes, that's correct. So 54 base 10 is 204 base 5. Therefore, the quotient of 413_5 divided by 2_5 is 204_5.

But hold on, maybe I should try doing the division directly in base 5 to confirm? Sometimes converting back and forth can introduce errors, especially if I made a mistake in conversion. Let me try that method as well to verify.

So, let's perform the division 413_5 ÷ 2_5 directly in base 5. Let me recall how division works in different bases. It's similar to base 10, but all the operations are done in base 5.

First, let me write down the dividend and divisor:

Dividend: 413_5

Divisor: 2_5

So, we need to divide 413_5 by 2_5. Let me set it up like long division.

First, I need to see how many times 2_5 goes into the first digit of the dividend. The first digit is 4_5. Let me convert 4_5 to base 10 to understand its value. 4 in base 5 is 4 in base 10. So 4 divided by 2 is 2. So 2_5 goes into 4_5 twice. So I write 2 as the first digit of the quotient.

Multiply 2_5 by 2_5. Let me compute that in base 5. 2*2 is 4, so 4_5. Subtract that from 4_5, which gives 0. Bring down the next digit, which is 1_5. So now we have 01_5.

Now, how many times does 2_5 go into 1_5? Well, 2_5 is 2 in base 10, and 1_5 is 1 in base 10. So 2 goes into 1 zero times. So we write 0 as the next digit in the quotient. Multiply 0 by 2_5, which is 0, subtract that from 1_5, giving 1_5. Bring down the next digit, which is 3_5. So now we have 13_5.

Now, how many times does 2_5 go into 13_5? Let me convert 13_5 to base 10 to make it easier. 1*5 + 3 = 8. And 2_5 is 2 in base 10. So 8 divided by 2 is 4. So 4 times. So we write 4 as the next digit in the quotient. Multiply 4_5 by 2_5. Let me compute that in base 5. 4*2 is 8 in base 10. Now, convert 8 to base 5. 5 goes into 8 once with a remainder of 3. So 8 is 13_5. Therefore, 4_5 * 2_5 = 13_5. Subtract that from 13_5, which gives 0. So there's no remainder.

Therefore, the quotient is 2 0 4_5, which is 204_5. That matches the result I got earlier when converting to base 10. So that seems to confirm that the answer is indeed 204_5.

But just to be thorough, let me check once more by converting 204_5 back to base 10 and see if it equals 54. So 2*5^2 + 0*5^1 + 4*5^0 = 2*25 + 0 + 4 = 50 + 4 = 54. Which is correct. And 54 divided by 2 is 27? Wait, no, wait. Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? Wait, no, 54 divided by 2 is 27? 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Solution:
We may carry out long division in base 5 just as in base 10. We have  \[
\begin{array}{c|ccc}
\multicolumn{2}{r}{2} & 0 & 4 \\
\cline{2-4}
2 & 4 & 1 & 3 \\
\multicolumn{2}{r}{4} & \downarrow & \\ \cline{2-2}
\multicolumn{2}{r}{0} & 1 & \\
\multicolumn{2}{r}{} & 0 & \downarrow \\ \cline{3-3}
\multicolumn{2}{r}{} & 1 & 3 \\
\multicolumn{2}{r}{} & 1 & 3 \\ \cline{3-4}
\multicolumn{2}{r}{} & & 0
\end{array}
\]for a quotient of $\boxed{204_5}$. Note that in the above calculation we have used that $13_5$ divided by $2_5$ is $4_5$, which follows from $4_5\times2_5=8_{10}=13_5$.