### Instruction: A base-10 integer $n=4\triangle_9=\triangle0_7$, where $\triangle$ represents a single digit. What is the value of $n$?### Response: I need to find a base-10 integer that satisfies two different representations in base-9 and base-7.
One way to do this is to convert both representations to base-10 and set them equal to each other.
To convert $4\triangle_9$ to base-10, I need to multiply the digits by powers of 9, starting from the rightmost one.
So $4\triangle_9=4\times 9^1+\triangle\times 9^0=36+\triangle$.
Similarly, to convert $\triangle0_7$ to base-10, I need to multiply the digits by powers of 7, starting from the rightmost one.
So $\triangle0_7=\triangle\times 7^1+0\times 7^0=7\triangle$.
Now I can set them equal to each other and solve for $\triangle$.
$36+\triangle=7\triangle\implies 6\triangle=36\implies \triangle=6$.
Now that I have $\triangle=6$, I can plug it back into either representation and get the value of $n$.
I'll use $4\triangle_9=4\times 9^1+6\times 9^0=36+6=42$.
So $n=42$.