First, let us convert $629_{10}$ to each of the two bases. To convert to base 7, we realize $7^{4}>629_{10}>7^{3}$. So, we can tell that $629_{10}$ in base seven will have four digits. $7^{3}=343$, which can go into 629 only one time at most, leaving $629-1\cdot343 = 286$ for the next three digits. $7^{2}=49$ goes into 286 five times at most, leaving us with $286-5\cdot49 = 41$. Then, $7^{1}=7$ goes into 41 five times at most, leaving $41-5\cdot7 = 6$ for the ones digit. All together, the base seven equivalent of $629_{10}$ is $1556_{7}$.

To convert to base 8, we realize similarly that $8^{4}>629_{10}>8^{3}$. So, we can tell that $629_{10}$ in base eight will have four digits. $8^{3}=512$, which can go into 629 only one time at most, leaving $629-1\cdot512 = 117$ for the next three digits. $8^{2}=64$ goes into 117 one time at most, leaving us with $117-1\cdot64 = 53$. Then, $8^{1}=8$ goes into 53 six times at most, leaving $53-6\cdot8 = 5$ for the ones digit. All together, the base eight equivalent of $629_{10}$ is $1165_{8}$.

Finally, comparing $1556_{7}$ and $1165_{8}$, we find that digits 1, 5, and 6 are present in both numbers.  Thus, there are $\boxed{3}$ digits in common.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
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.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
3
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.
test