### Instruction: Find $325_{8}-237_{8}$. Express your answer in base $8$.### Response: I know that base $8$ means that each digit represents a power of $8$, so for example, $325_{8}=3\cdot 8^2+2\cdot 8^1+5\cdot 8^0=192+16+5=213_{10}$ in base $10$.
Similarly, $237_{8}=2\cdot 8^2+3\cdot 8^1+7\cdot 8^0=128+24+7=159_{10}$.
I could convert both numbers to base $10$, subtract them, and then convert the result back to base $8$, but that would be tedious and prone to errors.
Instead, I will try to do the subtraction directly in base $8$.
To do that, I need to align the digits by their place values and subtract them column by column, starting from the rightmost one.
If I run into a situation where the top digit is smaller than the bottom digit, I need to borrow one from the next higher place value and add $8$ to the top digit.
For example, $5-7$ in base $8$ is not $-2$, but $5+8-7=6$ with a borrow of $1$ from the next column.
So, let me start with the rightmost column: $5-7$.
I need to borrow one from the next column, so I mark a $-1$ above the $2$ and add $8$ to the $5$.
Now I have $13-7=6$ in base $8$, so I write $6$ below the column.
Moving to the next column, I have $2-3$, but I need to subtract the borrowed $1$ as well, so it is actually $1-3$.
Again, I need to borrow one from the next column, so I mark a $-1$ above the $3$ and add $8$ to the $1$.
Now I have $9-3=6$ in base $8$, so I write $6$ below the column.
Finally, moving to the leftmost column, I have $3-2$, but I need to subtract the borrowed $1$ as well, so it is actually $2-2$.
[* 8 This is just $0$ in base $8$, so I write $0$ below the column.
I have completed the subtraction, and the result is $66_{8}$.
I can check this by converting it to base $10$ and comparing it with the difference of the base $10$ representations of the original numbers.
$66_{8}=6\cdot 8^1+6\cdot 8^0=48+6=54_{10}$.
$213_{10}-159_{10}=54_{10}$, so it matches.
Therefore, I am confident that the answer is correct.