<Question>
How many of the same digits are found in the base 7 and base 8 representations of $629_{10}$? For example, $121_{3}$ and $413_{5}$ would have one digit in common.
</Question>

<think>
<Plan_1>
Okay, let's see. The problem is asking how many of the same digits are found in the base 7 and base 8 representations of the decimal number 629. For example, if we have numbers like 121 in base 3 and 413 in base 5, they share the digit 1, so the answer would be 1. Got it.

First, I need to figure out what 629 in decimal is when converted to base 7 and base 8. Then, I have to compare the digits in those two base representations and count how many digits are the same.

Let me start with converting 629 from base 10 to base 7.

To convert a decimal number to another base, the standard method is to divide the number by the base and keep track of the remainders. The remainders, read in reverse order, give the representation in the new base.

So, let's do that for base 7.

<prompt_1.1> Follow the above plan to convert 629 from base 10 to base 7. </prompt_1.1>

Now, let's convert 629 to base 8.

Again, using the same division by base and remainder method.

<prompt_1.2> Follow the above plan to convert 629 from base 10 to base 8. </prompt_1.2>
</Plan_1>

<Execution_1>
<execution_1.1>
Dividing 629 by 7. Let's see, 7*80 = 560. 629 - 560 = 69. So, 7*80 = 560, remainder 69. But wait, that's not the right way. Wait, actually, the process is to divide the number by 7, get the quotient and remainder. Then divide the quotient by 7 again, and so on, until the quotient is 0. Then the remainders, read in reverse, give the base 7 number.

Let me do that step by step.

First division: 629 ÷ 7.

7*80 = 560. 629 - 560 = 69. So, 629 = 7*80 + 69. But wait, that can't be right because 7*80 is 560, but 560 + 69 is 629. However, the remainder should be less than 7. Wait, 69 divided by 7 is 9 with a remainder of 6. So, actually, 629 ÷ 7 is 89 with a remainder of 6. Let me check: 7*89 = 623. 629 - 623 = 6. Yes, that's correct. So the first remainder is 6.

Now, take 89 and divide by 7. 7*12 = 84. 89 - 84 = 5. So, remainder is 5. So second remainder is 5.

Take 12 and divide by 7. 7*1 = 7. 12 - 7 = 5. So remainder is 5. Third remainder is 5.

Take 1 and divide by 7. 0 quotient with remainder 1. So fourth remainder is 1.

Now, since the quotient is 0, we stop. So the remainders, in reverse order, are 1, 5, 5, 6. Therefore, 629 in base 7 is 1 5 5 6, which is 1556_7.
</execution_1.1>

<execution_1.2>
So, 629 divided by 8. Let's compute:

8*78 = 624, remainder 5. So the least significant digit is 5. Then 78 divided by 8 is 9, remainder 6. Next digit is 6. Then 9 divided by 8 is 1, remainder 1. Next digit is 1. Then 1 divided by 8 is 0, remainder 1. So the digits are 1, 1, 6, 5. Therefore, 629 in base 8 is 1165.
</execution_1.2>
</Execution_1>

<Plan_2>
Based on execution_1.1, execution_1.2:
Wait, let me check that again.

Wait, converting 629 to base 8:

<prompt_2.1> Verify the conversion of 629 to base 8. </prompt_2.1>

Based on execution_1.1:
So, now we have the two representations:

<prompt_2.2> List the digits of the base 7 and base 8 representations for comparison. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
First step: 629 ÷ 8. 8*70 = 560. 629 - 560 = 69. 69 ÷ 8 = 8*8=64, remainder 5. So quotient is 78 (since 8*78=624, which is 629 -5), so first remainder is 5. Then 78 ÷ 8 = 9 with remainder 6 (since 8*9=72, 78-72=6). Then 9 ÷ 8 = 1 with remainder 1. Then 1 ÷ 8 = 0 with remainder 1. So the remainders are 5, 6, 1, 1. But when we read them in reverse order, it's 1, 1, 6, 5. Therefore, 1165_8. Let me verify: 1*8^3 + 1*8^2 + 6*8 + 5 = 512 + 64 + 48 + 5 = 512+64=576, 576+48=624, 624+5=629. Correct. So base 8 is 1165.
</execution_2.1>

<execution_2.2>
Base 7: 1556_7 (digits 1, 5, 5, 6)
Base 8: 1165_8 (digits 1, 1, 6, 5)

Now, we need to compare these digits. Let's list them out:

Base 7 digits: 1, 5, 5, 6
Base 8 digits: 1, 1, 6, 5
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.2:
Looking for digits that appear in both. Let's go through each digit in base 7 and check if it's in base 8.

<prompt_3.1> Compare the digits of the base 7 and base 8 representations and count the common digits. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
Base 7 has digits 1, 5, 5, 6.

1: appears in base 8 (twice, but we just need to know if it's present). So that's one common digit.

5: appears in base 8 as well. So that's another common digit.

6: appears in base 8 too. So that's a third common digit.

Wait, but hold on. The digits in base 7 are 1, 5, 5, 6. So the unique digits are 1, 5, 6. Similarly, base 8 digits are 1, 1, 6, 5. Unique digits are 1, 6, 5. So all three digits 1, 5, 6 are present in both bases. Therefore, the number of common digits is 3.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Is that right?

Wait, but let me check again. Maybe I made a mistake in the conversion.

<prompt_4.1> Verify the solution by comparing the digits again. </prompt_4.1>

Based on execution_3.1:
Wait, but let me check if there's any other digits that might be common. For example, if a digit appears multiple times, but in this case, the digits in base 7 are 1,5,5,6.

<prompt_4.2> Re-evaluate the common digits to ensure no duplicates are counted. </prompt_4.2>

Based on execution_3.1:
Wait, but hold on. Let me check the problem statement again.

<prompt_4.3> Reconfirm the solution by reviewing the problem statement. </prompt_4.3>

Based on execution_3.1:
But let me just cross-verify. Let's list all digits in base 7: 1,5,5,6.

<prompt_4.4> Cross-verify the solution by listing and comparing the digits again. </prompt_4.4>
</Plan_4>

<Execution_4>
<execution_4.1>
Converting 629 to base 7: Let's do it step by step again.

629 divided by 7: 7*89=623, remainder 6. So first digit (rightmost) is 6.

89 divided by 7: 7*12=84, remainder 5. Next digit is 5.

12 divided by 7: 7*1=7, remainder 5. Next digit is 5.

1 divided by 7: 0, remainder 1. Leftmost digit is 1. So 1556_7. Correct.

Base 8: 629 divided by 8: 8*78=624, remainder 5. Rightmost digit 5.

78 divided by 8: 8*9=72, remainder 6. Next digit 6.

9 divided by 8: 1*8=8, remainder 1. Next digit 1.

1 divided by 8: 0, remainder 1. Leftmost digit 1. So 1165_8. Correct.

So digits in base 7: 1,5,5,6. Base 8: 1,1,6,5. So overlapping digits are 1,5,6. Each appears at least once in both. So three digits.
</execution_4.1>

<execution_4.2>
So in terms of unique digits, they are the same. So 1,5,6. So three digits. But wait, the problem says "how many of the same digits are found". So if a digit appears multiple times in both, does each occurrence count? Wait, no, the question is about the digits found in both representations. So if a digit is present in both, regardless of how many times, it's counted once. Wait, but the example given is 121_3 and 413_5. They share the digit 1, so the answer is 1. So it's the count of distinct digits that are common. So in this case, the common digits are 1,5,6. So three digits. Therefore, the answer is 3.
</execution_4.2>

<execution_4.3>
It says "how many of the same digits are found in the base 7 and base 8 representations". So, for example, if a digit appears multiple times in both, does each occurrence count? Wait, the example given is 121 base 3 and 413 base 5, which have one digit in common (the 1). So even though 1 appears twice in the first number and once in the second, they count the number of shared digits, not the number of shared occurrences. So in the example, 121 and 413 share the digit 1, so that's one common digit, even though 1 appears twice in the first number. Therefore, in our problem, we need to find the set of digits that are present in both base 7 and base 8 representations, regardless of how many times they appear. So if both representations have a 1, 5, and 6, then three digits. Therefore, the answer is 3.
</execution_4.3>

<execution_4.4>
So digits present: 1,5,6.

Digits in base 8: 1,1,6,5. So digits present: 1,6,5. So intersection is {1,5,6}, which has three elements. Therefore, three digits in common.

But wait, the example given in the problem is 121_3 and 413_5, which have one digit in common (the 1). So in that case, even though 1 appears twice in the first number, it's still counted once. So here, we need to count the number of digits that are present in both, regardless of how many times they appear. So if both have at least one of the same digit, it counts as one.

But in our case, all three digits 1,5,6 are present in both. Therefore, the answer is 3.

But wait, let me check again the base 7 and base 8 representations.

Base 7: 1556_7. So digits are 1,5,5,6.

Base 8: 1165_8. So digits are 1,1,6,5.

So the digits present in both are 1,5,6. So three digits. Therefore, the answer is 3.
</execution_4.4>
</Execution_4>

<Plan_5>
Based on execution_4.4:
But let me confirm once more.

<prompt_5.1> Confirm the solution by reviewing the entire process. </prompt_5.1>
</Plan_5>

<Execution_5>
<execution_5.1>
Maybe there's a digit that I missed. Let's list all digits in base 7: 1,5,5,6. So digits are 1,5,6. In base 8: 1,1,6,5. So digits 1,6,5. So intersection is 1,5,6. Therefore, three digits. So the answer should be 3.

But let me check if there's a digit that I might have miscounted.

Wait, another way: list all digits in base 7 and see if they are in base 8.

Digits in base 7: 1,5,5,6.

1 is in base 8.

5 is in base 8.

5 is in base 8.

6 is in base 8.

So all four digits (but with duplicates) are in base 8. But since we are counting unique digits, it's three.

Alternatively, if we think of the problem as counting how many digits are common between the two numbers, considering multiplicity, but the example shows that even with multiple occurrences, it's counted once. So the answer is 3.

Therefore, I think the answer is 3. So the final answer is 3, boxed.

**Final Answer**
</execution_5.1>
</Execution_5>

<Final_answer>
Therefore, there are $\boxed{3}$ digits that appear in both the base $7$ and base $8$ representations of $629_{10}$.
</Final_answer>
</think>